INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS. = u (ξ), ξ (0, t).
|
|
- Felicia Taylor
- 6 years ago
- Views:
Transcription
1 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS 0. Background and history The mean value theorem gives, for u C 0 (R + ), u(t) t = u(t) u(0) t 0 Hence you might suspect that an inequality of the type 0 u p dt c p 0 = u (ξ), ξ (0, t). t p dt, c p > 0 would by likely to hold for these u. This is indeed true and this is of course the famous original Hardy inequality. The story of this inequality goes back to the 920 s and to the person who it is named after, G.Hardy. Hardy himself proved the corresponding integral version of the inequality i.e. ( ) p p ( x p f(t) p dt f(t)dt) dx p x 0 for p >, f(t) 0, where f is integrable over every finite interval (0, A) and the constant in front of the integral is the best possible. In this thesis, however, we will henceforth only consider the differential versions of this inequality and it s generalizations. It is hard to give a precise definition of what a Hardy (or Hardy-type) inequality is, because of the large number of possibilities of generalizing it. A sufficently precise definition, covering many cases, is given for example in [29] : Definition (Hardy s inequality in R n ) ( ) ( u(x) p p v(x)dx C with R n and where w(x), v(x) are weight functions and 0 < q, p. 0 0 ) u(x) q q w(x)dx Of course, not all such inequalities are true for some C > 0. Neccesary and sufficent conditions for such inequalities to hold in one dimension has been extensively studied by many different authors during the last fifty years. Such conditions usually involves integral criteria for the weight functions w and v.
2 2 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS One of the main reasons to study L 2 Hardy inequalities is their close connections to the Schrödinger equation u = u + V (x)u, = Planck s constant i t and especially the special case that descibes the stationary states u V (x)u = 0 which is one of the cornerstones of classical quantum mechanics. Also, the L p inequalities are of great importance in the study of the p-laplacian and the p-schrödinger equation div ( u p 2 u ) V (x, u) = 0 and it s generalizations. In this thesis we will give examples of applications of Hardy inequalities (from now on abbreviated H.I.) to Schrödinger operators describing many particle systems. Also, in article 4, we will see how improved H.I. may be a valuable tool in the theory of viscous incompressible flow. This is a slight generalization and improvement of previous work by P. Sobolevskii (see [4]). Except for that, the main focus in this thesis will not be on applications, but more on the mathematical aspects of the theory. I will below give a short summary over recent achivements in the theory of Hardy inequalities, which are similar in fashion to my results. The text will not give a complete picture of the field, but it will exemplify with some of the highlights. The interest in Hardy-type inequalities seems to have grown enormously in the last decades, especially in the last ten years lots of important papers have been published. The total material available concerning Hardy, and related, inequalities are extensive. Some references, for the interested reader, may be found in the bibliography at the end of this introduction. 0.. Different aspects and generalizations of Hardy s inequality. The classical generalization of H.I. to higher dimensions is p (0.) u p dx n p Rn R p x dx, u n p C 0 (R n \ {0}), where the constant is the best possible. For p = 2 this inequality is also called the uncertanity principle. The proof of (0.) can be found for example in the, nowadays almost classical, book Hardy type Inequalities by B.Opic and A.Kufner [24]. Another really classical book, which has been a source of inspiration for numerous people through the years, worth mentioning here is Inequalities by G.Hardy, J.E.Littlewood and G.Pólya [24]. A more modern book which has been very influential and rewarding for many researchers in the theory of Hardy-type inequalities, including myself, is Sobolev Spaces by V.Maz ya [37]. Also, for a modern introduction to the theory of H.I., especially in the one dimensional case, the book [30] is recommendable.
3 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS 3 In two dimensions the uncertainity principle does not hold, but if we weaken the singularity a bit by adding a logarithmic term or/and some extra conditions to the functions u, one can get some results. For example, it is true that R2 u 2 u 2 dx C R 2 x 2 ( + ln 2 x ) dx, if u(x)dx = 0 x = and R2 u 2 u 2 dx C R x dx, if u(x)dx = 0 r > x =r (see [43]). For bounded domains and n 3 we have that p u p dx n p,p p dx u W x p 0 () for n > p, where the constant is optimal. That is, the boundedness does not change the value of the best constant. There also exists higher order analogues of the above inequalites, which are applicable to the corresponding operators of higher order, for example the bi-harmonic operator. Examples of such inequalities are the Rellich s inequality [42]: Theorem[Rellich] Let n > 4 and be a bounded domain in R n, then u 2 dx n2 (n 4) 4 u 2 2,2 dx, u W 6 x 4 0 (). We also have a generalization of Rellich inequality to the L p case due to E.B.Davies and A.M.Hinz (see [5]): Theorem[Davies,Hinz] ( ) p (n 2p)(p )n (0.2) u p dx p 2 2,p dx, u W x 2p 0 (), n > 2p. For even higher derivatives, the below theorem was proved in the same article: Theorem[Davies,Hinz] Let m N 0, d N. i) If < p < and n > 2mp, then (0.3) m u p dx C (m, p, n) dx, x 2mp 2m,p u W0 (), where ( C (m, p, n) = m p (n 2kp)(2(k )p + (p )n)). p 2mp k= ii) If p < and n > (2m + )p, then (0.4) ( m u) p dx C 2 (m, p, n) dx, x u (2m+)p W 2m+,p (),
4 4 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS where C 2 (m, p, n) = ( m p (n p)p (n (2k + )p)((2k )p + (p )d)). p (2m+)p k= 0.2. Fractional derivatives. Another way to generalize the H.I. is to consider fractional(non-integer) derivatives also. These are much harder to handle than the ordinary derivatives because the non-local definition of them. In L 2, these may be defined using the Fourier transform. In general one can define the fractional Sobolev space of functions, W s,p 0 (), in some domain, by the completion of C0 () in the seminorm ( ) u(x) u(y) p /p dxdy (p ) (0 < s < ). x y n+sp One of the first multidimensional Hardy inequalities involvning fractional norms was proven by T.Kato [27]: Theorem[Kato] (u, u) 2 R3 u 2 π x dx, u C 0 (R 3 ). The constant 2/π here cannot be improved. This inequality has important consequences in the theory of stability of matter. In a paper [48] from 2000, D.Yafaev obtained the best constants in the higher order and fractional order generalizations of (0.) for functions from W l,2 0 () : Theorem[Yafaev] Let u W l,2 (R n ), then C l ξ R 2l u(ξ) 2 dξ x 2l û(x) 2 dx if l < n/2 n R n and C l ξ R 2l u(ξ) 2 dξ x 2l u(x) (α!) ( α u)(0)x α 2 dx n R n if l > n/2, The best constants here are given by If l is an integer this gives α m l n/2 / Z and m = [l n/2]. C l = 2 2l max{ Γ2 ((n/2 l)/2) Γ 2 ((n/2 + l)/2), Γ2 ((n/2 l + )/2) Γ 2 ((n/2 + l + )/2) }. C l = 2 2l (n 2l) 2 (n 2l + 4) 2... (n + 2l 4) 2. Yafaev used decomposition of the functions u into spherical harmonics to reduce the problem to a one dimensional inequality which is proved by diagonalizing a certain integral operator with the help of the Mellin transform.
5 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS 5 Various results have since then been published concerning inequalites involving fractional Sobolev norms and L p generalizations of the above result. One such result is by V.Maz ya and T. Shaposhnikova: Theorem[Maz ya,shaposhnikova] Let n, p, 0 < s < and sp < n. Then, for an arbitrary function u W s,p 0 (R n ), there holds s( s) Rn u(x) u(y) p Rn c(n, p) dx. (n sp) p x y n+sp x sp R n The main point of this theorem was to show a conjecture of J.Bourgain, H.Brezis and P.Mironescu about the asymptotic behaviour of the constant on the RHS as s tends to 0, or n/p (see [7],[8]). Their proof is completely elementary, but very clever. The value of c(n, p) given in the article is not proven to be optimal. I do not know if the optimal value has been found yet or not. It would certainly be interesting to find out. An older influential article about sharp constant in similar Hardy-Sobolev type inequalities is [33] by E.Lieb Improved optimal Hardy inequalities. One step further, when the best possible constant in an inequality has been found, is to try to see if one could add positive reminder terms containing singularities of lower order. This is often possible in the Hardy case. Many different types of remainder terms are possible. Of course, the possibilities depend very much on the weight function in the main term. One way to improve the H.I. (0.) and it s generalizations (0.3),(0.4) is to add remainder terms of the same type as the main term, but with different exponents. This has been done, for example in the article [2], by F.Gazzola, H.C.Grunau and E.Mitidieri. They proved the following result : Theorem[Gazzola,Grunau,Mitidieri] Let R n be a sufficently smooth domain and let k N, 2k n. Then there exist constants c,..., c k depending only on k, n and, such that for every u W 2,k () with j u = 0 for j N 0 and 2j < k there holds: ( m u 2 dx 2m + 4(m j)) 4 j=(n 2) 2m if k = 2m, m N and m u 2 dx if k = 2m +, m N 0. + ( 2m+ 4 2m+ 2m+ l= c l j= u 2 x dx + 2m c 4m l l= (n + 4m + 2 4j) 2 ) u 2 x 4m+2 2l dx, u 2 dx x 4m 2l u 2 dx x 4m+2
6 6 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS The techniques used here are also a based on reduction of dimension. An infinite series improvent of (0.) involving iterated logartimic weights can be found in [9]: Theorem[Filippas,Tertikas] Let X (t) = ( ln t), X k (t) = X (X k (t)), k = 2, 3,... and let D sup x x. Then, the following inequality hold for any u W,2 0 (): ( ) 2 n 2 u 2 u 2 dx n x dx x 2 X2 ( x D )X2 2( x D )... X2 i ( x D ) u 2 dx. i= Morover, for each k =, 2,... the constant /4 is the best constant for the corresponding k-improved Hardy inequality, that is 4 = inf u 2 dx ( ) n 2 2 u 2 dx k n x 2 4 i= X x X Xi 2 u 2 dx u W,2. 0 () X 2 x X Xk 2 u 2 dx Later, this inequality has been generalized to the L p case and more general weights [4]. Another interesting type of remainder terms, which improve inequality (0.) were obtained in [2]: Theorem[Gazzola,Grunau,Mitidieri] Let n > p, then ( ) n p u p C(p, n) dx dx + p x p p n dx, u W,p 0 (). (The special case with p = 2 is due to H.Brezis and J.L.Vázquez [0]). We will see later how this type of remainder term appear naturally in other Hardy type inequalities as well. Another similar L p inequality, which improves (0.2), is Theorem[Gazzola,Grunau,Mitidieri] Let p 2, let R n (n 2p) be a bounded domain and let e n = B(0, ) = V ol(b(0, )). Then for any u W 2,p () W,p 0 () one has: ( ) p (n 2p)(p )n u p a(p, n) b(p, n) dx dx + dx + dx. p 2 x 2p 2/n x 2p 2 2p/n One natural question here is whether the optimal fractional order inequalities, for examples the ones by Yafaev described above (see [48]), allows one to add reminder terms similar to the ones in the integer order case.
7 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS Improved optimal Hardy inequalities involvning distance functions. In my research I have focused on inequalities where the distance function to the origin, appearing in the main term in the classical Hardy inequality, have been replaced by the distance function to the boundary, δ(x). Those inequalities depend very much on the geometrical properties of. A basic question here is to find the largest possible constant, C p, in the inequality u p dx C p δ(x) dx, u p C 0 () for a specific class of domains R n. For simply connected domains there exists no such positive constant, not depending on the dimension [3]. However, in 986, A. Ancona proved that C 2 for 6 R2 []. It is still an open question whether this constant is optimal or not. One class of domains that are particulary nice and bit easier to treat here are the convex ones. For this subclass the best constant is equal to (p ) p /p p regardless of the dimension. This inequality was proven in the -dimensional case by Hardy himself in 925 [22],[23]. The 2-dimensional case was proven in 997 by T. Matskewich and P.E.Sobolevskii and they pointed out that their proof also holds in any dimension [40]. The proof is based on the fact that it is sufficent to prove the inequality for convex polytopes. Another proof of the inequality in it s full generality(for all p > and all dimensions) was given the following year in an appendix of an article by M.Marcus, V.J.Mizel and Y.Pinchover [35]. Since then a lot of results concerning generalizations and improvements of this type of inequalities have been published by many different authors. Much focus has been given to the question of what type of remainder terms one can obtain. Especially the problem of finding the largest possible constant, D p,n (), such that u p dx C p,n () δ(x) p dx + D p,n() where C p,n () is the optimal constant for the main term i.e. C p,n () = inf u p dx u C0 () dx, δ(x) p dx, u C 0 (), R 2, has been given much attention (see for example [4],[5],[6], [9],[20],[26],[37], [44] and [45]). The ideal situation here would be to know the optimal value of D p,n () for every specific choice of. That is in most cases too much to hope for, so one has to satisfy with some lower(and upper) estimates of the constant. Usually such estimates may be expressed in terms of some geometric properties of. Examples of such estimates for convex domains are : where C p,n () α(p, n) (diam()) p, C p,n() β(p, n), C p p,n () n γ(p, n) δ in () p, diam() = sup x y, = Vol() and δ in () = sup δ(x) x,y x The proofs can be found in [9](in the case p = 2), [26] and [44] respectively.
8 8 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS One may note that first type of estimate follows from the second one which in turn follows from the third one. Another important field of study here is about the question of obtaining as large potential, V (x), as possible such that u p dx C p,n () δ(x) dx + V (x) dx p holds. Here significant contributions have been given by several authors. For example G.Barbatis, S.Filippas and A.Tertikas have been given much insights into this problem in their work (see [4],[5], [6],[20]). One of their main theorems will be given below after some preliminaries: Define recursively X (t) = ( ln t), t (0, ), X k (t) = X (X k (t)), k = 2, 3,... Let be a domain in R n and K a piecewise smooth surface of codimension k. Furthermore, let δ(x) = dist(x, K) fulfil the conditions (0.5) p δ p k p and Then we have the following theorem : = div( δ p k p p 2 δ p k p ) 0 sup δ(x) <. x in \ K Theorem[Barbatis,Filippas,Tertikas] There exists a positive constant D 0 = D 0 (k, p) sup x δ(x) such that for any D D 0 and all u W,p 0 ( \ K) there holds p u p dx k p p δ dx p + p k p ( p 2 ) 2p p δ p X2 ( δ D )... X2 i ( δ D )dx If 2 p < k, then we can take D 0 = sup x δ(x) Moreover, the constants in front of each terms are the best possible in each of the following cases : (a) (b) (c) i= k = n and K = {0} k = and K = 2 k n and K One may note (as in Tertikas) that if p = 2, K = and is convex, then condition (0.5) is automatically satisfied. Hardy type inequalities involvning distance functions will be discussed further in the summaries of my articles.
9 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS Further applications of Hardy s inequality in physics. Except for the direct application of Hardy s inequality to the Schrödinger operator, other useful variants has been successfully developed for applications in other areas of physics. I will give some examples of this below. J.Dolbeault, M.J.Esteban and E.Séré proved an optimal Hardy-type inequality involving Spinors and Pauli matrices [7]: Theorem[Dolbeault,Esteban,Séré] Let σ = (σ i ) i=,2,3 be the Pauli-matrices: ( ) ( ) ( ) 0 0 i 0 σ =, σ 0 2 =, σ i 0 3 = 0 Then for all u W,2 (R 3, C 2 ). R 3 ( σ u 2 + x ) R3 + u 2 u 2 dx x dx, Several extensions and generalizations of this result may be found in the article [8]. The proofs in the mentioned article are very nice examples of clever applications of ideas from [4],[5], [6]. A very interesting and popular subject of study is Hardy-type inequalities related to the magnetic Schrödinger operator and its corresponding magnetic form h[u, u] = ( i a) 2 dx u C0(R n ), n 2, R n where a is a vector of real-valued functions belonging to L 2 (R n ) In a very popular article A.Laptev and T.Weidl [32] showed among other things that an inequality of the type R2 (i + a)u 2 u 2 dx C R + x dx, u 2 2 C 0 (R 2 ), holds where the constant C depends strongly on the magnetic field. They also showed that in some cases also the H.I. R2 (i + a)u 2 u 2 dx C R x dx, u 2 2 C 0 (R 2 \ {0}) hold for some C > 0. This is in great constrast to the ordinary H.I. (with a = 0) which does not hold in R 2. Examples of magnetic fields which gives rise to such an inequality are those of so-called Aharonov-Bohm type (see [32]). In this case it was proved, in the same article, that the optimal constant C takes the form C = min Φ n 2 n Z where Φ = B(x)dx, B = curl a 2π denotes the total flux of the magnetic field B = curl a.
10 0 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS Generalizations of the above results may be found, for example, in [2] or [3].
11 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS Article - A geometrical version of Hardy s inequality for W,p () In a very influential paper [9] H.Brezis and M.Marcus showed that the following improved Hardy inequality holds: Theorem[Brezis,Marcus] Let be convex, then (0.6) u 2 dx 4 u 2 dx + λ() δ 2 (x) u 2 dx, u W,2 0 (), where λ() 4 diam 2 (). They also formulated the question whether the constant λ() could be replaced by a constant of the form D() = γ 2 n or not, where γ is some universal constant. This was proven to be true by M. and T.Hoffmann-Ostenhof and A.Laptev in the following theorem (see [26]): Theorem[M. and T.Hoffmann-Ostenhof,Laptev] Let be convex, then (0.7) where u 2 dx 4 u 2 δ 2 (x) dx + µ n 2 n u 2 dx, u W,2 0 (), µ n = n(n 2)/n S n 2/n 4 They also obtained some refinements involvning logarithmic weights. In my article I generalize this result to the L p case. More precisely, I prove the following inequality : Theorem[Tidblom] Let be convex, then ( ) p p (0.8) u p a(p, n) dx dx + p δ p (x) p n dx, u W,p 0 () The volume dependent constants in front of the remainder terms in (0.7),(0.8) are superior to the inverse diameter constant in (0.6) if the domain in question has a large diameter but small volume, for example if the domain is long and thin. I also show how to obtain similar results for non convex domains. In that case, however, the constant in front of the main term is not likely to be optimal. The method I use is a variant of the elegant, elementary and geometrical method used in [26]. The constant in front of the remainder terms in (0.7),(0.8) is probably far from being optimal. No one has so far being able to find the optimal value. Barbatis, Filippas and Tertikas showed that if is a ball and p = 2, n = 3 then the best constant C n = inf u W,2 0 () B u 2 dx 4 B u 2 dx B u 2 δ 2 dx
12 2 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS is equal to the first eigenvalue, µ of the Dirichlet Laplacian for the unit disc in R 2. For arbitrary n 2 they obtained the estimate (n )(n 3) C n µ Recently, Filippas, Tertikas and Mazya generalized and improved my result by showing that Theorem[Filippas,Tertikas,Maz ya] Suppose R n is a convex domain with δ in <. For < p < n and p q < np, let C() be the best constant in the inequality n p ( ) p ( ) p u p p dx dx + C() u q q dx p δp for C() of the form a(p, n) C() = δ p where δ in = sup δ(x) = the in-radius. in x Then there exist positive constants c i = c i (p, q, n), i =, 2 independent of, such that np n p q c (p, q, n)δin np n p q C() c 2 (p, q, n)δin. This constant is of a stronger type than mine that involved the volume of, since we always have an estimate B n (0, ) δ n in. There are even a lot of examples of domains with infinte volume, but finite in-radius. A simple example of such a domain is an infinite cylinder. In those cases the inverse volume constant become zero whereas the inverse in-radius constant is nonzero. A natural open question left for the future is to see if one can find an even better type of geometric depending constant than the one involvning the inradius.
13 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS 3 Article 2 - A Hardy inequality in the Half-space Consider an unbounded convex domain,, in R n. One might wonder what type of remainder terms one can add to the optimal Hardy inequality : ( ) p p u p dx p δ dx, p C 0 () in this case. The remainder terms obtained in [20],[26] and [44] of the form C() dx where C() = a(p, n) p n resp. C() = b(p, n) δ p in are usually not very good here, since the first one is always equal zero and the other one might be zero too. Let us consider one of the simplest examples of an unbounded domain, namely the upper half-space in R n, R n +. Every one dimensional linear Hardy inequality on the positive real axis (0.9) 0 ( ) p p u p dt p 0 t p dt + 0 v(t) dt immediately give rise to a corresponding inequality in R n + simply by changing t to x n and integrating over all the other variables. Hovever, it is known that you cannot have an inequality like (0.9) where v(t) is nonnegative a.e. Therefore, you have to do something else if you want a positive remainder term. The first improved optimal Hardy inequality in the litterature, is precisely about this situation. The proof is due to V.G.Mazya and appears in his book Sobolev spaces : Theorem[Maz ya] (0.0) u 2 dx 4 R n + R n + He begins by proving the inequality u 2 dx + x 2 n 6 R n + u 2 dx, u C x n (x 2 n + x 2 n) 0 (R n +). 2 Theorem[Maz ya] (0.) x n p u p dx Rn dx, u C R (2p) n p (x 2 n + x 2 n) 0 (R n ) 2 and then he substitutes u(x) = x n /2 v(x) in (0.), which leads to inequality (0.0) after some integrations by parts. By using elementary methods, based on a vector field approach as in [6], I will show how to improve the constant /6 in (0.0) to /8.
14 4 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS I will also prove an old conjecture by Mazya which says that there should exist positive constants C(p, τ) > 0 such that ( ) p p u p dx dx + C(p, τ) R n p + R n δp + R n x p τ dx. + n (x 2 n + x 2 τ/2 n) The same method can be used prove other inequalities for unbounded domains as well.
15 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS 5 Article 3 - Geometric many particle Hardy inequalities In this article we study Hardy inequalities related to Schrödinger operators describing many particle systems. In R 3N we show that N C(N) x i x j 2 in the quadratic form sense, where the x k are points in R 3 and C(N) = 4( + K(N)), where K(N) will be defined below. Also, the asymptotical behaviour of K(N) is investigated. We prove that K(N) N 2, 2 and hence that the optimal choice of C(N) satisfies C(N) 2N. Furthermore, we prove that K(N) lim inf N N > δ > 0. In particular we get that K(N) = O(N) when N. The problem of finding the sharp value of K(N) is directly related to the problem of finding the optimal configuration of points which maximizes the quotient Q N (x,..., x N ) = i j N i j,i k,j k 2 N i j where r ij = x i x j and R ijk is the circumradius of the triangle obtained by letting it s corners to be the points x i, x j, x k. In fact K(N) is defined as the supremum of this quotient. For three points the optimal configuration turns out to be the equilateral traingle and for four points the tetrahedron is optimal. For larger values of N, the problem is much harder. This problem of finding the right configuration of points gives a nice connection between the theory of Hardy inequalities and Geometrical combinatorics. r 2 ij R 2 ijk,
16 6 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS Article 4 - Various results in the theory of Hardy Inequalities Here we give more examples of H.I. that can be proved with the same type of techniques used in my previous articles. I will, among other things, prove that, in the upper half-space in R n the following inequality holds true : n u 2 u 2 dx 4 x 2 n k dx, u C x2 0 (R n +) n R n + k=0 R n + where the constant /4 in front of each term is optimal. In the domain = {x,..., x n } I will prove the inequality u 2 dx ( ) u 2 dx + (n ) 2 u 2 dx, 4 x 2 x 2 n x x 2 n where (n ) 2 also is optimal. This will be used to prove that u 2 dx ( ) u 2 dx 4 x 2 x 2 n ( n ) + c(n) u 2 dx, x k (x x 2 n) 2 where (n ) 2 c(n) = 2 n(2n + n 2). This c(n) is however most likely not optimal. k= In later sections I will make some comments of how to generalize and improve some known inequalities. As an example, I will generalize my results in article to include higher order derivatives and vector valued functions (in the one dimensional case). I will also calculate an explict expression for the largest possible angle, α, such that u 2 dx u α 4 α 2 δ(x) dx, u 2 C 0 (), where α is a conical sector of angle α in R 2 and δ(x) = dist(x, α ). This has been done numerically by E.B. Davies in [3] and the result up to three decimals was that α The exact value is ( α = π + 4 arctan 4 Γ ( ) Γ ( 4 ) 2 ) ( = π + 4 arctan Some questions about similar problems will also be disdussed. In the last section, we will see how optimal H.I. with remainder terms can be used to prove existence and uniqueness in certain special cases of the stationary Navier-Stokes equations { ν v + n k= v k v x k = p + f(x) div v = 0, ).
17 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS 7 where f(x) is a known vector valued function and we seek v(x) and p(x). References [] A.Ancona, On strong barriers and an inequality by Hardy for domains i R N, London Math. Soc. (2) 34 (986), pp [2] A.Balinsky, Hardy type inequalities for Aharonov-Bohm potentials with multiple singularities, Math. Res. Lett. 0, (2003), pp. 8. [3] A.Balinsky, A.Laptev and V.Sobolev, Generalized Hardy inequality for the Magnetic Dirichlet forms, Journal of Statistical Physics 6, (2004), pp [4] G.Barbatis, S.Filippas and A.Tertikas, Series expansion for L p Hardy inequalities, Indiana Univ. Math. J. 52 No. (2003), pp [5] G.Barbatis, S.Filippas and A.Tertikas, Refined geometric L p Hardy inequalities, Comm. Contemp. Math. 5 (6) (2003), pp [6] G.Barbatis, S.Filippas and A.Tertikas, A unified approach to improved L p Hardy inequalities with best constants, Transactions of the AMS 356 (2004), pp [7] J.Bourgain, H.Brezis, P.Mironescu, Another look at Sobolev spaces, Optimal Control and P.D.E. (J. L. Menaldi, E. Rofman and A. Sulem, eds.), a volume in honour of A. Bensoussan s 60th birthday, IOS Press (200), pp [8] J.Bourgain, H.Brezis, P.Mironescu, Limiting embedding theorems for W ɛ,p (R n ), Journal d Analyse 87, (2002), pp [9] H.Brezis and M.Marcus, Hardy s inequalities revisited, Dedicated to Ennio De Giorgi. Ann.Scuola Norm. Sup.Pisa Cl.Sci (4) vol. 25 (997) no. 2 (998), pp [0] H.Brezis and J.L Vasquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complutense Madrid 0 (997), pp [] F.Colins and Y.Hupperts, Minimization problems related to generalized Hardys inequalities, Nonlinear analysis 52 (2003) pp [2] E.B.Davies, Heat kernels and spectral theory, [Cambridge, Cambridge University press,989]. [3] E.B.Davies, The Hardy constant, Quart.J.math. Oxford (2),vol. 46, (995) pp [4] E.B.Davies, Spectral theory and differential operators, Cambridge studies in advanced mathematics vol. 42, [Cambridge, Cambridge University Press, 995]. [5] E.B.Davies, A.M.Hinz, Explicit constants for Rellich inequalities in L p (), Math. Zeitschrift, vol. 227, (998) pp [6] E.B.Davies, A review of Hardy inequalities, Oper. Theory Adv. Appl.,vol. 0, (998) pp [7] J.Dolbeault, M.J.Esteban and E.Séré, On the eigenvalues of operators with gaps. Applications to Dirac operators., Journal of Func. Anal.(2), vol. 74 (), (2000) pp [8] J.Dolbeault, M.J.Esteban, M.Loss and L.Vega, An analytical proof of Hardy-like inequalities related to the Dirac operator, Journal of Func. Anal., vol. 26 (), (2004) pp. 2. [9] S.Filippas and A.Tertikas, Optimizing Improved Hardy Inequalities, Jornal of func. Anal. 92, (2002), pp [20] S.Filippas, V.Maz ya and A.Tertikas, On a question of Brezis and Marcus, preprint [2] F.Gazzola, H-C.Grunau and E.Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. of the A.M.S. 356 (2004), pp [22] G.Hardy, Note on a theorem of Hilbert, Math. Zeitschr.(6),(920) pp [23] G.Hardy, An inequality between integrals, Messenger of Math. 54,(925) pp [24] G.Hardy, J.E.Littlewood and G.Pólya, Inequalities, second edition, Cambridge Mathematical Library, [Cambridge, Cambridge University Press, 952]. [25] I.W.Herbst, Spectral theory of the operator (p 2 + m 2 ) /2 Ze 2 /r, Comm. Math. Phys. 53 (977), pp [26] M.Hoffmann Ostenhof,T.Hoffmann Ostenhof and A.Laptev, A geometrical version of Hardy s inequality, J.Func.Anal. 89 (2002), pp [27] T.Kato, Perturbation Theory for Linear Operators, Springer Verlag, Berlin, 966. [28] A.Kufner and B.Opic, Hardy type inequalities, Pitman Research notes in mathematics series vol. 29, [London, Longman Group UK Limited, 990].
18 8 INTRODUCTION AND A SHORT OVERVIEW OF RECENT RESULTS [29] A.Kufner and Lars-Erik Persson, The Hardy Inequality. Research Report , Dep. of mathematics, Luleå university of technology. [30] A.Kufner and Lars-Erik Persson, Weighted Inequalities of Hardy Type, World Scientific, [3] O.A.Ladyzhenskaya, Mathematical theory of viscous incompressible Flow, Gordon and Breach Science Publishers, New york 969. [32] A.Laptev and T.Weidl, Hardy inequalities for magnetic Dirichlet forms, Operator Theory: Advances and Applications vol. 08 (999) pp , Birkhäuser verlag, Basel/Switzerland. [33] E.H.Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related Inequalities, Annals of math. vol. 8 (983) pp [34] E.H.Lieb and M.Loss, Analysis, Graduate Studies in Mathematics vol. 4, [USA, American Mathematical Society 997]. [35] M.Marcus, V.J.Mizel and Y.Pinchover, On the best constant for Hardys inequality in R n, Trans. of the A.M.S. vol. 350 (998) pp [36] T.Matskewich and P.E.Sobolevskii, The best possible constant in a generalized Hardy s inequality for convex domains in R n, Nonlinear Anal.vol. 28 (997) pp [37] V.Mazýa, Sobolev Spaces, [Berlin, Springer Verlag 985]. [38] V.Mazýa, T.Shaposhnikova On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, Journal de Mathmatiques Pures et Appliques, vol. 8 (2002) pp [39] V.Mazýa, T.Shaposhnikova On the Brezis and Mironescu conjecture concerning a Gagliardo-Nirenberg inequality for fractional Sobolev norms, Journal of Functional Anal.vol. 28 (997) pp [40] T.Matskewich, P.E.Sobolevskii, The best possible constant in a generalized Hardys inequality for convex domains in R n, Nonlinear Anal. TMA, vol. 28 (997) pp [4] P.E.Sobolevskii, Hardy s Inequality for the Stokes Problem, Nonlinear Anal. vol. 30, No., (997) pp [42] F.Rellich, Halbbeschränkte Differentialoperatoren höheren Ordnung,in: J.C.H.Gerresten et al.(eds.), Proceedings of the International Congress of Mathematicians Amsterdam 954, Vol III, pp , Groningen: Nordhoff, 956. [43] M.Z.Solomyak, A Remark on the Hardy Inequalities, Integr. Equat. Oper. Th. 9 (994), pp [44] J.Tidblom, A geometrical version of Hardy s inequality for W,p (), Proc. AMS. 32 (8) (2004) pp [45] J.Tidblom, A Hardy inequality in the half-space, Journal of Functional Analysis. 22 (2005) pp [46] A.Wannebo, Hardy inequalities, Proc. of the A.M.S. 09 () (990) pp [47] A.Wannebo, Hardy inequalities and imbeddings in domains generalizing C 0,λ domains, Proc. of the A.M.S. 22 (4) (994) pp [48] D. Yafaev, Sharp Constants in the Hardy-Rellich Inequalities, Journal of Functional Analysis. 68 (999) pp Jesper Tidblom, Department of Mathematics, Stockholm University, 06 9 Stockholm, Sweden. E mail : jespert@math.su.se
On the Brezis and Mironescu conjecture concerning a Gagliardo-Nirenberg inequality for fractional Sobolev norms
On the Brezis and Mironescu conjecture concerning a Gagliardo-Nirenberg inequality for fractional Sobolev norms Vladimir Maz ya Tatyana Shaposhnikova Abstract We prove the Gagliardo-Nirenberg type inequality
More informationOn the structure of Hardy Sobolev Maz ya inequalities
J. Eur. Math. Soc., 65 85 c European Mathematical Society 2009 Stathis Filippas Achilles Tertikas Jesper Tidblom On the structure of Hardy Sobolev Maz ya inequalities Received October, 2007 and in revised
More informationSHARP L p WEIGHTED SOBOLEV INEQUALITIES
Annales de l Institut de Fourier (3) 45 (995), 6. SHARP L p WEIGHTED SOBOLEV INEUALITIES Carlos Pérez Departmento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid, Spain e mail: cperezmo@ccuam3.sdi.uam.es
More informationHARDY INEQUALITIES WITH BOUNDARY TERMS. x 2 dx u 2 dx. (1.2) u 2 = u 2 dx.
Electronic Journal of Differential Equations, Vol. 003(003), No. 3, pp. 1 8. ISSN: 107-6691. UL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) HADY INEQUALITIES
More informationTHE L 2 -HODGE THEORY AND REPRESENTATION ON R n
THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 167 176 ON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES Piotr Haj lasz and Jani Onninen Warsaw University, Institute of Mathematics
More informationThe Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge
The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge Vladimir Kozlov (Linköping University, Sweden) 2010 joint work with A.Nazarov Lu t u a ij
More informationTOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS. Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017
TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017 Abstracts of the talks Spectral stability under removal of small capacity
More informationPólya-Szegö s Principle for Nonlocal Functionals
International Journal of Mathematical Analysis Vol. 12, 218, no. 5, 245-25 HIKARI Ltd, www.m-hikari.com https://doi.org/1.12988/ijma.218.8327 Pólya-Szegö s Principle for Nonlocal Functionals Tiziano Granucci
More informationAN EXAMPLE OF FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W 1,p FOR EVERY p > 2 BUT NOT ON H0
AN EXAMPLE OF FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W,p FOR EVERY p > BUT NOT ON H FERNANDO FARRONI, RAFFAELLA GIOVA AND FRANÇOIS MURAT Abstract. In this note we give an example of functional
More informationTHE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION. Juha Kinnunen. 1 f(y) dy, B(x, r) B(x,r)
Appeared in Israel J. Math. 00 (997), 7 24 THE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION Juha Kinnunen Abstract. We prove that the Hardy Littlewood maximal operator is bounded in the Sobolev
More informationMathematical Research Letters 4, (1997) HARDY S INEQUALITIES FOR SOBOLEV FUNCTIONS. Juha Kinnunen and Olli Martio
Mathematical Research Letters 4, 489 500 1997) HARDY S INEQUALITIES FOR SOBOLEV FUNCTIONS Juha Kinnunen and Olli Martio Abstract. The fractional maximal function of the gradient gives a pointwise interpretation
More informationA Product Property of Sobolev Spaces with Application to Elliptic Estimates
Rend. Sem. Mat. Univ. Padova Manoscritto in corso di stampa pervenuto il 23 luglio 2012 accettato l 1 ottobre 2012 A Product Property of Sobolev Spaces with Application to Elliptic Estimates by Henry C.
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More informationINSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES. Note on the fast decay property of steady Navier-Stokes flows in the whole space
INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES Note on the fast decay property of stea Navier-Stokes flows in the whole space Tomoyuki Nakatsuka Preprint No. 15-017 PRAHA 017 Note on the fast
More informationBLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED
BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis
More informationASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS. Tian Ma. Shouhong Wang
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 11, Number 1, July 004 pp. 189 04 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS Tian Ma Department of
More informationWeighted norm inequalities for singular integral operators
Weighted norm inequalities for singular integral operators C. Pérez Journal of the London mathematical society 49 (994), 296 308. Departmento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid,
More informationLORENTZ SPACE ESTIMATES FOR VECTOR FIELDS WITH DIVERGENCE AND CURL IN HARDY SPACES
- TAMKANG JOURNAL OF MATHEMATICS Volume 47, Number 2, 249-260, June 2016 doi:10.5556/j.tkjm.47.2016.1932 This paper is available online at http://journals.math.tku.edu.tw/index.php/tkjm/pages/view/onlinefirst
More informationCourse Description for Real Analysis, Math 156
Course Description for Real Analysis, Math 156 In this class, we will study elliptic PDE, Fourier analysis, and dispersive PDE. Here is a quick summary of the topics will study study. They re described
More informationON SOME ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS
Chin. Ann. Math.??B(?), 200?, 1 20 DOI: 10.1007/s11401-007-0001-x ON SOME ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS Michel CHIPOT Abstract We present a method allowing to obtain existence of a solution for
More informationDETERMINATION OF THE BLOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION
DETERMINATION OF THE LOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION y FRANK MERLE and HATEM ZAAG Abstract. In this paper, we find the optimal blow-up rate for the semilinear wave equation with a power nonlinearity.
More informationABOUT SOME TYPES OF BOUNDARY VALUE PROBLEMS WITH INTERFACES
13 Kragujevac J. Math. 3 27) 13 26. ABOUT SOME TYPES OF BOUNDARY VALUE PROBLEMS WITH INTERFACES Boško S. Jovanović University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11 Belgrade, Serbia
More information(b) If f L p (R), with 1 < p, then Mf L p (R) and. Mf L p (R) C(p) f L p (R) with C(p) depending only on p.
Lecture 3: Carleson Measures via Harmonic Analysis Much of the argument from this section is taken from the book by Garnett, []. The interested reader can also see variants of this argument in the book
More informationOn the Hardy constant of non-convex planar domains: the case of the quadrilateral
On the Hardy constant of non-convex planar domains: the case of the quadrilateral Gerassimos Barbatis University of Athens joint work with Achilles Tertikas University of Crete Heraklion, Hardy constant
More informationNonlinear aspects of Calderón-Zygmund theory
Ancona, June 7 2011 Overture: The standard CZ theory Consider the model case u = f in R n Overture: The standard CZ theory Consider the model case u = f in R n Then f L q implies D 2 u L q 1 < q < with
More informationA LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE. 1.
A LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE THOMAS CHEN AND NATAŠA PAVLOVIĆ Abstract. We prove a Beale-Kato-Majda criterion
More informationProceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005
Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWO-DIMENSIONAL SCHRÖDINGER
More informationNew Perspectives. Functional Inequalities: and New Applications. Nassif Ghoussoub Amir Moradifam. Monographs. Surveys and
Mathematical Surveys and Monographs Volume 187 Functional Inequalities: New Perspectives and New Applications Nassif Ghoussoub Amir Moradifam American Mathematical Society Providence, Rhode Island Contents
More informationLOGARITHMIC SOBOLEV INEQUALITY REVISITED
LOGARITHMIC SOBOLEV IEQUALITY REVISITED HOAI-MIH GUYE AD MARCO SQUASSIA Abstract. We provide a new characterization of the logarithmic Sobolev inequality. 1. Introduction The classical Sobolev inequality
More informationIntegro-differential equations: Regularity theory and Pohozaev identities
Integro-differential equations: Regularity theory and Pohozaev identities Xavier Ros Oton Departament Matemàtica Aplicada I, Universitat Politècnica de Catalunya PhD Thesis Advisor: Xavier Cabré Xavier
More informationGlobal regularity of a modified Navier-Stokes equation
Global regularity of a modified Navier-Stokes equation Tobias Grafke, Rainer Grauer and Thomas C. Sideris Institut für Theoretische Physik I, Ruhr-Universität Bochum, Germany Department of Mathematics,
More informationSpectral theory of first order elliptic systems
Spectral theory of first order elliptic systems Dmitri Vassiliev (University College London) 24 May 2013 Conference Complex Analysis & Dynamical Systems VI Nahariya, Israel 1 Typical problem in my subject
More informationON THE HÖLDER CONTINUITY OF MATRIX FUNCTIONS FOR NORMAL MATRICES
Volume 10 (2009), Issue 4, Article 91, 5 pp. ON THE HÖLDER CONTINUITY O MATRIX UNCTIONS OR NORMAL MATRICES THOMAS P. WIHLER MATHEMATICS INSTITUTE UNIVERSITY O BERN SIDLERSTRASSE 5, CH-3012 BERN SWITZERLAND.
More informationResearch Statement. Xiangjin Xu. 1. My thesis work
Research Statement Xiangjin Xu My main research interest is twofold. First I am interested in Harmonic Analysis on manifolds. More precisely, in my thesis, I studied the L estimates and gradient estimates
More informationMinimal periods of semilinear evolution equations with Lipschitz nonlinearity
Minimal periods of semilinear evolution equations with Lipschitz nonlinearity James C. Robinson a Alejandro Vidal-López b a Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. b Departamento
More informationFree boundaries in fractional filtration equations
Free boundaries in fractional filtration equations Fernando Quirós Universidad Autónoma de Madrid Joint work with Arturo de Pablo, Ana Rodríguez and Juan Luis Vázquez International Conference on Free Boundary
More informationMinimization problems on the Hardy-Sobolev inequality
manuscript No. (will be inserted by the editor) Minimization problems on the Hardy-Sobolev inequality Masato Hashizume Received: date / Accepted: date Abstract We study minimization problems on Hardy-Sobolev
More informationThe Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:
Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply
More informationSUBELLIPTIC CORDES ESTIMATES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 000-9939XX0000-0 SUBELLIPTIC CORDES ESTIMATES Abstract. We prove Cordes type estimates for subelliptic linear partial
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationFractional Sobolev spaces with variable exponents and fractional p(x)-laplacians
Electronic Journal of Qualitative Theory of Differential Equations 217, No. 76, 1 1; https://doi.org/1.14232/ejqtde.217.1.76 www.math.u-szeged.hu/ejqtde/ Fractional Sobolev spaces with variable exponents
More informationFRACTIONAL SOBOLEV SPACES WITH VARIABLE EXPONENTS AND FRACTIONAL P (X)-LAPLACIANS. 1. Introduction
FRACTIONAL SOBOLEV SPACES WITH VARIABLE EXPONENTS AND FRACTIONAL P (X-LAPLACIANS URIEL KAUFMANN, JULIO D. ROSSI AND RAUL VIDAL Abstract. In this article we extend the Sobolev spaces with variable exponents
More informationClassification of Solutions for an Integral Equation
Classification of Solutions for an Integral Equation Wenxiong Chen Congming Li Biao Ou Abstract Let n be a positive integer and let 0 < α < n. Consider the integral equation u(x) = R n x y u(y)(n+α)/()
More informationPartial Differential Equations, 2nd Edition, L.C.Evans Chapter 5 Sobolev Spaces
Partial Differential Equations, nd Edition, L.C.Evans Chapter 5 Sobolev Spaces Shih-Hsin Chen, Yung-Hsiang Huang 7.8.3 Abstract In these exercises always denote an open set of with smooth boundary. As
More informationSELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS
More informationExistence of minimizers for the pure displacement problem in nonlinear elasticity
Existence of minimizers for the pure displacement problem in nonlinear elasticity Cristinel Mardare Université Pierre et Marie Curie - Paris 6, Laboratoire Jacques-Louis Lions, Paris, F-75005 France Abstract
More informationarxiv: v1 [math.ap] 28 Mar 2014
GROUNDSTATES OF NONLINEAR CHOQUARD EQUATIONS: HARDY-LITTLEWOOD-SOBOLEV CRITICAL EXPONENT VITALY MOROZ AND JEAN VAN SCHAFTINGEN arxiv:1403.7414v1 [math.ap] 28 Mar 2014 Abstract. We consider nonlinear Choquard
More informationSEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT
Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC
More informationA generalised Ladyzhenskaya inequality and a coupled parabolic-elliptic problem
A generalised Ladyzhenskaya inequality and a coupled parabolic-elliptic problem Dave McCormick joint work with James Robinson and José Rodrigo Mathematics and Statistics Centre for Doctoral Training University
More informationOn a compactness criteria for multidimensional Hardy type operator in p-convex Banach function spaces
Caspian Journal of Applied Mathematics, Economics and Ecology V. 1, No 1, 2013, July ISSN 1560-4055 On a compactness criteria for multidimensional Hardy type operator in p-convex Banach function spaces
More informationTakens embedding theorem for infinite-dimensional dynamical systems
Takens embedding theorem for infinite-dimensional dynamical systems James C. Robinson Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. E-mail: jcr@maths.warwick.ac.uk Abstract. Takens
More informationThe Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria
ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-19 Wien, Austria The Negative Discrete Spectrum of a Class of Two{Dimentional Schrodinger Operators with Magnetic
More informationNECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES
NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES JUHA LEHRBÄCK Abstract. We establish necessary conditions for domains Ω R n which admit the pointwise (p, β)-hardy inequality u(x) Cd Ω(x)
More informationSome Remarks About the Density of Smooth Functions in Weighted Sobolev Spaces
Journal of Convex nalysis Volume 1 (1994), No. 2, 135 142 Some Remarks bout the Density of Smooth Functions in Weighted Sobolev Spaces Valeria Chiadò Piat Dipartimento di Matematica, Politecnico di Torino,
More informationUniversität des Saarlandes. Fachrichtung 6.1 Mathematik
Universität des Saarlandes U N I V E R S I T A S S A R A V I E N I S S Fachrichtung 6.1 Mathematik Preprint Nr. 155 A posteriori error estimates for stationary slow flows of power-law fluids Michael Bildhauer,
More informationSPECTRAL PROBLEMS IN SPACES OF CONSTANT CURVATURE
131 SPECTRAL PROBLEMS IN SPACES OF CONSTANT CURVATURE RAFAEL D. BENGURIA Departamento de Física, P. Universidad Católica de Chile, Casilla 306, Santiago 22, CHILE E-mail: rbenguri@fis.puc.cl Here, recent
More informationMAIN ARTICLES. In present paper we consider the Neumann problem for the operator equation
Volume 14, 2010 1 MAIN ARTICLES THE NEUMANN PROBLEM FOR A DEGENERATE DIFFERENTIAL OPERATOR EQUATION Liparit Tepoyan Yerevan State University, Faculty of mathematics and mechanics Abstract. We consider
More information2 A Model, Harmonic Map, Problem
ELLIPTIC SYSTEMS JOHN E. HUTCHINSON Department of Mathematics School of Mathematical Sciences, A.N.U. 1 Introduction Elliptic equations model the behaviour of scalar quantities u, such as temperature or
More informationSINC PACK, and Separation of Variables
SINC PACK, and Separation of Variables Frank Stenger Abstract This talk consists of a proof of part of Stenger s SINC-PACK computer package (an approx. 400-page tutorial + about 250 Matlab programs) that
More informationSobolev Spaces. Chapter Hölder spaces
Chapter 2 Sobolev Spaces Sobolev spaces turn out often to be the proper setting in which to apply ideas of functional analysis to get information concerning partial differential equations. Here, we collect
More informationA global solution curve for a class of free boundary value problems arising in plasma physics
A global solution curve for a class of free boundary value problems arising in plasma physics Philip Korman epartment of Mathematical Sciences University of Cincinnati Cincinnati Ohio 4522-0025 Abstract
More informationMULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN
Electronic Journal of Differential Equations, Vol. 016 (016), No. 97, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIONS
More informationConstrained Leja points and the numerical solution of the constrained energy problem
Journal of Computational and Applied Mathematics 131 (2001) 427 444 www.elsevier.nl/locate/cam Constrained Leja points and the numerical solution of the constrained energy problem Dan I. Coroian, Peter
More informationTitle: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on
Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on manifolds. Author: Ognjen Milatovic Department Address: Department
More informationESTIMATES FOR ELLIPTIC HOMOGENIZATION PROBLEMS IN NONSMOOTH DOMAINS. Zhongwei Shen
W,p ESTIMATES FOR ELLIPTIC HOMOGENIZATION PROBLEMS IN NONSMOOTH DOMAINS Zhongwei Shen Abstract. Let L = div`a` x, > be a family of second order elliptic operators with real, symmetric coefficients on a
More informationarxiv: v1 [math.ap] 16 Jan 2015
Three positive solutions of a nonlinear Dirichlet problem with competing power nonlinearities Vladimir Lubyshev January 19, 2015 arxiv:1501.03870v1 [math.ap] 16 Jan 2015 Abstract This paper studies a nonlinear
More informationNONHOMOGENEOUS ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT AND WEIGHT
Electronic Journal of Differential Equations, Vol. 016 (016), No. 08, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONHOMOGENEOUS ELLIPTIC
More informationGAKUTO International Series
1 GAKUTO International Series Mathematical Sciences and Applications, Vol.**(****) xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx, pp. xxx-xxx NAVIER-STOKES SPACE TIME DECAY REVISITED In memory of Tetsuro Miyakawa,
More informationJUHA KINNUNEN. Harmonic Analysis
JUHA KINNUNEN Harmonic Analysis Department of Mathematics and Systems Analysis, Aalto University 27 Contents Calderón-Zygmund decomposition. Dyadic subcubes of a cube.........................2 Dyadic cubes
More informationNONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian
More informationThe Brézis-Nirenberg Result for the Fractional Elliptic Problem with Singular Potential
arxiv:1705.08387v1 [math.ap] 23 May 2017 The Brézis-Nirenberg Result for the Fractional Elliptic Problem with Singular Potential Lingyu Jin, Lang Li and Shaomei Fang Department of Mathematics, South China
More informationGRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS
LE MATEMATICHE Vol. LI (1996) Fasc. II, pp. 335347 GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS CARLO SBORDONE Dedicated to Professor Francesco Guglielmino on his 7th birthday W
More informationarxiv: v1 [math.ca] 15 Dec 2016
L p MAPPING PROPERTIES FOR NONLOCAL SCHRÖDINGER OPERATORS WITH CERTAIN POTENTIAL arxiv:62.0744v [math.ca] 5 Dec 206 WOOCHEOL CHOI AND YONG-CHEOL KIM Abstract. In this paper, we consider nonlocal Schrödinger
More informationGlobal well-posedness for semi-linear Wave and Schrödinger equations. Slim Ibrahim
Global well-posedness for semi-linear Wave and Schrödinger equations Slim Ibrahim McMaster University, Hamilton ON University of Calgary, April 27th, 2006 1 1 Introduction Nonlinear Wave equation: ( 2
More informationBIHARMONIC WAVE MAPS INTO SPHERES
BIHARMONIC WAVE MAPS INTO SPHERES SEBASTIAN HERR, TOBIAS LAMM, AND ROLAND SCHNAUBELT Abstract. A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed.
More informationMULTIPLE SOLUTIONS FOR AN INDEFINITE KIRCHHOFF-TYPE EQUATION WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 2015 (2015), o. 274, pp. 1 9. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIOS
More informationTOPICS. P. Lax, Functional Analysis, Wiley-Interscience, New York, Basic Function Theory in multiply connected domains.
TOPICS Besicovich covering lemma. E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, N.J., 1971. Theorems of Carethedory Toeplitz, Bochner,...
More informationLiouville-type theorems and decay estimates for solutions to higher order elliptic equations
Liouville-type theorems decay estimates for solutions to higher order elliptic equations Guozhen Lu, Peiyong Wang Jiuyi Zhu Abstract. Liouville-type theorems are powerful tools in partial differential
More informationTHE HOT SPOTS CONJECTURE FOR NEARLY CIRCULAR PLANAR CONVEX DOMAINS
THE HOT SPOTS CONJECTURE FOR NEARLY CIRCULAR PLANAR CONVEX DOMAINS YASUHITO MIYAMOTO Abstract. We prove the hot spots conjecture of J. Rauch in the case that the domain Ω is a planar convex domain satisfying
More informationSome lecture notes for Math 6050E: PDEs, Fall 2016
Some lecture notes for Math 65E: PDEs, Fall 216 Tianling Jin December 1, 216 1 Variational methods We discuss an example of the use of variational methods in obtaining existence of solutions. Theorem 1.1.
More informationA survey on l 2 decoupling
A survey on l 2 decoupling Po-Lam Yung 1 The Chinese University of Hong Kong January 31, 2018 1 Research partially supported by HKRGC grant 14313716, and by CUHK direct grants 4053220, 4441563 Introduction
More informationON THE BEHAVIOR OF THE SOLUTION OF THE WAVE EQUATION. 1. Introduction. = u. x 2 j
ON THE BEHAVIO OF THE SOLUTION OF THE WAVE EQUATION HENDA GUNAWAN AND WONO SETYA BUDHI Abstract. We shall here study some properties of the Laplace operator through its imaginary powers, and apply the
More informationHARNACK S INEQUALITY FOR GENERAL SOLUTIONS WITH NONSTANDARD GROWTH
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 37, 2012, 571 577 HARNACK S INEQUALITY FOR GENERAL SOLUTIONS WITH NONSTANDARD GROWTH Olli Toivanen University of Eastern Finland, Department of
More informationRemarks on the blow-up criterion of the 3D Euler equations
Remarks on the blow-up criterion of the 3D Euler equations Dongho Chae Department of Mathematics Sungkyunkwan University Suwon 44-746, Korea e-mail : chae@skku.edu Abstract In this note we prove that the
More informationLecture Note III: Least-Squares Method
Lecture Note III: Least-Squares Method Zhiqiang Cai October 4, 004 In this chapter, we shall present least-squares methods for second-order scalar partial differential equations, elastic equations of solids,
More informationSplines which are piecewise solutions of polyharmonic equation
Splines which are piecewise solutions of polyharmonic equation Ognyan Kounchev March 25, 2006 Abstract This paper appeared in Proceedings of the Conference Curves and Surfaces, Chamonix, 1993 1 Introduction
More informationAuthor(s) Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 41(1)
Title On the stability of contact Navier-Stokes equations with discont free b Authors Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 4 Issue 4-3 Date Text Version publisher URL
More informationA dual form of the sharp Nash inequality and its weighted generalization
A dual form of the sharp Nash inequality and its weighted generalization Elliott Lieb Princeton University Joint work with Eric Carlen, arxiv: 1704.08720 Kato conference, University of Tokyo September
More informationA CLASS OF SCHUR MULTIPLIERS ON SOME QUASI-BANACH SPACES OF INFINITE MATRICES
A CLASS OF SCHUR MULTIPLIERS ON SOME QUASI-BANACH SPACES OF INFINITE MATRICES NICOLAE POPA Abstract In this paper we characterize the Schur multipliers of scalar type (see definition below) acting on scattered
More informationLIFE SPAN OF BLOW-UP SOLUTIONS FOR HIGHER-ORDER SEMILINEAR PARABOLIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 21(21), No. 17, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LIFE SPAN OF BLOW-UP
More informationThe De Giorgi-Nash-Moser Estimates
The De Giorgi-Nash-Moser Estimates We are going to discuss the the equation Lu D i (a ij (x)d j u) = 0 in B 4 R n. (1) The a ij, with i, j {1,..., n}, are functions on the ball B 4. Here and in the following
More informationInvariant measures and the soliton resolution conjecture
Invariant measures and the soliton resolution conjecture Stanford University The focusing nonlinear Schrödinger equation A complex-valued function u of two variables x and t, where x R d is the space variable
More informationAsymptotic behavior of infinity harmonic functions near an isolated singularity
Asymptotic behavior of infinity harmonic functions near an isolated singularity Ovidiu Savin, Changyou Wang, Yifeng Yu Abstract In this paper, we prove if n 2 x 0 is an isolated singularity of a nonegative
More informationA Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators
A Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators Lei Ni And I cherish more than anything else the Analogies, my most trustworthy masters. They know all the secrets of
More informationRESEARCH STATEMENT. Introduction
RESEARCH STATEMENT PRITHA CHAKRABORTY Introduction My primary research interests lie in complex analysis (in one variable), especially in complex-valued analytic function spaces and their applications
More informationVANISHING-CONCENTRATION-COMPACTNESS ALTERNATIVE FOR THE TRUDINGER-MOSER INEQUALITY IN R N
VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE FOR THE TRUDIGER-MOSER IEQUALITY I R Abstract. Let 2, a > 0 0 < b. Our aim is to clarify the influence of the constraint S a,b = { u W 1, (R ) u a + u b = 1 } on
More informationRegularity of the p-poisson equation in the plane
Regularity of the p-poisson equation in the plane Erik Lindgren Peter Lindqvist Department of Mathematical Sciences Norwegian University of Science and Technology NO-7491 Trondheim, Norway Abstract We
More information