2 BAISHENG YAN When L =,it is easily seen that the set K = coincides with the set of conformal matrices, that is, K = = fr j 0 R 2 SO(n)g: Weakly L-qu
|
|
- Jade Perkins
- 5 years ago
- Views:
Transcription
1 RELAXATION AND ATTAINMENT RESULTS FOR AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL BAISHENG YAN Abstract. Consider functional I(u) = R jjdujn ; L det Duj dx whose energy-well consists of matrices satisfying jj n = L det : We show that the relaxations of this functional in various Sobolev spaces are signicantly dierent. We also make several remarks concerning various p-growth semiconvex hulls of the energy-well set and prove an attainment result for a special Hamilton-Jacobi equation, jduj n = L det Du, in the so-called grand Sobolev space W q) ( R n )withq = nl L+ :. Introduction and main results Given L, consider function f L () = jj n ; L det on the space M nn of n n matrices and integral functional (.) I(u) = jf L (Du(x))j dx = jjdu(x)j n ; L det Du(x)j dx where u is a mapping from domain R n to R n and Du(x) is the Jacobi matrix of u. In general, jj denotes the operator norm of m n matrix 2 M mn dened by jj =max h2r n jhj= jhj: The absolute energy minimizers of I(u) can be characterized as mappings satisfying the Hamilton-Jacobi equation: (.2) Du(x) 2 L = f 2 M nn jjj n = L det g a:e: x 2 : In the terminology used for phase transition problems (see, e.g., [3, 9, 0, ]), the set L is the energy-well of energy functional I(u): Note that L is also the boundary of the so-called L-quasiconformal set K L dened by K L = f 2 M nn jjj n L det g:
2 2 BAISHENG YAN When L =,it is easily seen that the set K = coincides with the set of conformal matrices, that is, K = = fr j 0 R 2 SO(n)g: Weakly L-quasiregular mappings are the mappings u 2 W p loc ( Rn ) satisfying Du(x) 2 K L almost everywhere in (see Iwaniec [6]). Many regularity and stability properties of weakly quasiregular mappings have been studied in connection with the certain semi-convex hulls and the attainment result of the quasiconformal sets K L in Yan [3, 4, 5]. In this paper we study similar problems related to the set L : The natural space for I(u) is the Sobolev space W n ( R n ): In this note we are interested in minimization of functional I(u) in dierent Sobolev spaces W p ( R n ). For this purpose, we dene the p-growth relaxation functions of I(u) as follows: (.3) g p () = inf u2w p ( R n ) =x jj jf L (Du(x))j dx: Clearly the function g p () is non-decreasing with respect to p : For any given function f : M mn! R, we dene the quasiconvexication of f to be (.4) f # () = inf 2C 0 ( Rm ) f( + D(x)) dx jj 8 2 M mn where C 0 ( R m ) stands for all smooth functions with compact support in : A density argument easily shows that g = jf L j # and hence the function g is also quasiconvex in the sense of Morrey. Recall that, in general, a function f : M mn! R is said to be quasiconvex (in the sense of Morrey) if it satises (.5) jj f( + D(x)) dx f() 8 2 M mn 2 C 0 ( R m ): It is well-known that any nite quasiconvex function f is always rankone convex that is, f( + t) is convex in t 2 R for any 2 M mn with rank = : A function f : M mn! R is called polyconvex if it can be written as a convex function of all subdeterminants therefore all convex functions are polyconvex. It can be shown that a polyconvex function is always quasiconvex. We refer to Acerbi & Fusco [], Ball [2], Ball & Murat [4], Dacorogna [5], Morrey [8], and Muller [9] for the proofs and more properties of these semiconvex functions.
3 AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL 3 In what follows, we denote by f ; (0) the zero set of any scalar function f, that is, f ; (0) = f j f() =0g: We now state our main result. Theorem.. Let n 3 and L. Then (a) g p = jf L j # = maxff L 0g for all p n (b) for some >0, g p ; (0) = K L for all p n ; (c) g p 0 and the minimum is attained for all p< nl Part (c) of the theorem is equivalent to the following L+ : Theorem.2. Let n 3, L : Then, for any p< nl L+ and 2 M nn there exists a map u 2 W p ( R n ) with = x satisfying (.6) jdu(x)j n = L det Du(x) a:e: x 2 : Remark. This result sharpens an earlier result in Yan [4]. Note that, if = 0 the trivial map u 0 is a required solution of (.6) but, in this case, we can show existence of nontrivial solutions. In fact, we shall prove a sharper attainment result for a special class of nontrivial solutions of (.6) in a space strictly smaller than any W p ( R n ) for p< nl L+ : For a given q > we dene the grand Sobolev space W q) ( R m ) (see e.g. Iwaniec & Sbordone [7]) to be the space of all functions u 2\ p<q W p ( R m ) that satisfy [Du] q sup (q ; p) =p (.7) jdu(x)j dx p < : p<q jj Given any number 2 R, we dene the set (.8) S = fr(i +!!) j R 2 R 6= 0! 2 R n j!j =g where I is n n identity matrix and a b stands for the rank-one matrix (a i b j ). In the following, we shall use ( 2 n ) to denote the n n diagonal matrix with diagonal elements ::: n that is, we denote ( 2 n )= B A : 0 n Let J = I ; 2e e = (; ): Then J 2 = I and det J = ;: Dene JS = fjj 2 S g: Note that any matrix 2 S can be
4 4 BAISHENG YAN written as = Q(I +!!) for some >0, Q 2 SO(n) and! 2 R n with j!j = : Therefore, it follows easily that det = n ( + ) and jj = if j+j or jj = j+j if j+j : From this, one easily proves the following Proposition.3. For = L n; ; or L ; ;, S L for = ;L n; ; or ;L ; ;, JS L : In this paper, we shall also prove the following much stronger attainment result. Theorem.4. Let n 3 L be given. Then, for any 2 M nn, nl there exists a map u in the grand Sobolev space W L+ ) ( R n ) such that = x Du(x) 2 S [ JS 2 L a:e: x 2 for = L n; ; or L ; ; and 2 = ;L ; ; where the boundary condition = x is satised in the W p -sense for all p< nl L+ : Given any subset K of M mn and a number p, let C + p (K) be the set of continuous functions f : M mn! R satisfying 0 f() C(jj p +) fj K =0: Let QC + p (K), RC + p (K) andpc + p (K) be the class of quasiconvex, rankone convex, and polyconvex functions in C + p (K), respectively. Denition.. The p-growth quasiconvex, rank-one convex and polyconvex hulls of set K are dened, respectively, as follows: (.9) (.0) (.) Q p (K) =\ff ; (0) j f 2 QC + p (K)g R p (K) =\ff ; (0) j f 2 RC + p (K)g P p (K) =\ff ; (0) j f 2 PC + p (K)g: Remark. These p-growth semiconvex hulls have been referred to as the W p - or, simply, p-semiconvex hulls in Yan [2, 5]andYan & hou [6, 7], following the denition of usual semiconvex hulls without growth restriction given, for example, in Muller & Sverak [0] and Sverak []. We adopt the present denition here to stress the growth condition and to distinguish with the dierent and not equivalent denition of p-semiconvex hulls given in hang [8, 9]. The p-growth semiconvex hulls of the set K L have been studied in Yan [5]. Concerning the set L, we have the following
5 AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL 5 Theorem.5. A p-growth semiconvex hull of L is the same as that of K L. For example, R p ( L ) = M nn for p < nl L+ and R p( L ) = K L for p nl L+ moreover, Q p( L )=K L for p n ; with some >0: 2. Relaxations and semiconvex hulls In this section, we prove Theorem., parts (a) and (b), and Theorem.5. Part (c) of Theorem. and Theorem.2 follow from Theorem.4, which will be proved later in the paper. First of all,we observe the following useful result. Lemma 2.. lim jtj! f L ( + t) = + for any given 2 M nn with rank =: Proof. It suces to note that det( + t) =det + ct for a constant c if rank =, and hence as jtj!: f L ( + t) (jtjjj;jj) n ;jcjjtj;jdet j! Proof of Theorem.(a). Using a well-known relaxation principle for integral functionals (see, e.g., [, 5]) one can easily show that g n = jf L j # and hence, by denition, (2.) 0 g p g q jf L j # jf L j 8 p q: Since f L = jj n ;L det is W n -quasiconvex [4], we easily have g n f L and hence g n maxff L 0g: (This can be also derived directly from property of the determinant.) Therefore, by (2.), to prove Theorem.(a), it is sucient to show jf L j # = maxff L 0g: But this follows from Lemma 2. and the following general result: Proposition 2.2. Let f be a quasiconvex function. Suppose that for each 2 M nn there exists a rank-one matrix such that Then jfj # =maxff 0g: lim f( + t) =+: jtj!
6 6 BAISHENG YAN Proof. Obviously, since f is quasiconvex, jfj # maxff 0g: We prove the equality. Suppose, for the contrary, jfj # ( 0 ) > maxff( 0 ) 0g for some 0 : Since jfj # ( 0 ) jf( 0 )j we must have f( 0 ) < 0: Now consider h(t) =f( 0 + t 0 ) where 0 is a rank-one matrix such thath(t)! + as jtj!: Then, since h(0) = f( 0 ) < 0, we would have t < 0 < t 2 such that h(t ) = h(t 2 ) = 0: This implies jfj # ( 0 + t i 0 ) = 0 for i = 2: From this, since jfj # is rank-one convex, we would arrive at the conclusion that jfj # ( 0 ) 0, a contradiction. Therefore, jfj # maxff 0g: Proof of Theorem.(b). Let f = (maxff L 0g) =n : Then jf L j jf L j # = f n. Using Holder's inequality in denition (.3), it follows that, for p<n (2.2) [(f p ) # ] n=p g p jf L j # = f n : Since f is a homogeneous function of degree which vanishes exactly on the set K L and is also L n -mean coercive in the sense dened in Yan & hou [7], it follows from [7, Theorem 2.] that (f p ) # () = 0 if and only if 2 K L for p n ; with some >0 and therefore, by (2.2), g p () = 0 if and only if 2 K L : This proves the result. Proposition 2.3. Let g 0 be a rank-one convex function. If g() = 0 for all 2 L then g() =0for all 2 K L : Proof. Let 2 K L n L : Then f L () < 0: In a similar way as above, we have f L ( + t ) = f L ( + t 2 ) = 0 for some t < 0 < t 2 with a given rank-one matrix : Thus g( + t i ) = 0 for i = 2, which, by the rank-one convexity of g, implies g() = 0 and proves the result. Proof of Theorem.5. From denition of semiconvex hulls, to prove the theorem, it is sucient to prove the equalities: QC + p ( L )=QC + p (K L ) RC + p ( L )=RC + p (K L ) PC + p ( L )=PC + p (K L ): Since the functions in QC + p ( L ), RC + p ( L ), and PC + p ( L ) are all rankone convex, these equalities follow easily from Proposition 2.3. The proof is completed. 3. The lamination hull and attainment results Given any set K M mn and number p 2 [ ], dene p (K) to be the set of matrices 2 M mn such that there exists a map
7 AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL 7 u = u 2 W p ( R m ) satisfying (3.) Du(x) 2 K a:e: x 2 = x: Note that Lemma 3. in Yan [4] shows that the set p (K) is independent of the domain : From this denition, Theorem.2 is equivalent to the following Theorem 3.. p ( L )=M nn for all p< nl L+ : From this theorem, a similar argument as used for [4, Theorem 4.4] also derives the following shaper result we refer to [4] for details. Theorem 3.2. Let n 3, L and p < nl : Then, for any L+ piece-wise ane map ' 2 W p ( R n ) and > 0 there exists a map u 2 ' + W p 0 ( R n ) satisfying Du (x) 2 L L a.e. in and ku ; 'k L p () <: Let S be the set dened by (.8) in the introduction and dene a unbounded two-well set (3.2) W L = S [ JS 2 where 2 = ;L ; ; and is either L n; ; or L ; ; : Notice that Proposition.3 implies W L L therefore, Theorem 3. follows from an attainment result for set W L, which is a direct corollary of Theorem.4. Theorem 3.3. p (W L )=M nn for all p< nl L+ : To prove Theorem 3.3, or the more general Theorem.4, we need some techniques in convex integration theory we refer to [0,, 4] for more references on this theory. Denition 3.. Let L j (K) be dened for j = 0 2 inductively as follows: L 0 (K) =K and, for j =0 L j+ (K) =ft +(; t) t 2 [0 ] 2Lj (K) rank( ; ) g: The lamination hull of set K, L(K), is then dened to be (3.3) L(K) =[ j=0l j (K): The following useful attainment result has been given in Yan [4] see also Yan [3].
8 8 BAISHENG YAN Theorem 3.4 ([4, Theorem 3.2]). Let K M mn be a closed set and let A p (K) be a set satisfying (3.4) c 0 = sup jdu (x)j p dx < 2A jj where u 2 W p ( R n ) is some map satisfying (3:): Suppose the lamination hull B = L(A) is open and bounded. Then B p (K): Remarks. ) It follows from [4, Lemma 3.] and the remark following the proof of [4, Theorem 3.2] that, for any bounded domain R n and any 2 B = L(A), there exists a map u = u 2 W p ( R m ) such that (3.5) and (3.6) Du(x) 2 K a:e: x 2 = x jj jdu(x)j p dx maxfc 0 sup jj p g: 2B 2) A closer look at the proof of [4, Theorem 3.2] also shows that the map u = u satisfying (3.5), (3.6) above depends only on the family fu j 2 Ag in (3.4) and any xed number q p in particular, the solution u = u for any 2 B can be made independent of power p as long as p<qfor some q < : This can be seen from the choice of sequence f k g in the proof of [4, Theorem 3.2]. We notice that the construction of u = u depends on power p only through this sequence f k g that is required to satisfy k! 0 + and P k =p k < : However, if p<q, we can x such a sequence k satisfying k! 0 + and P k =q k < : Then the solution u = u constructed there is seen only depending on this sequence f k g and the family fu j 2 Ag given in (3.4). Note that the estimate (3.6) is independent of the sequence f k g. The main theorem of this section is the following Theorem 3.5. Let K M mn be a closed set and q > a number given. Suppose A M mn is a set such that for each 2 A there exists a map v = v 2 W q) (B R m ) satisfying Dv(x) 2 K a.e. x 2 B and = x where B is the unit ball in R n and suppose that (3.7) C q =sup[dv ] q B < : 2A If the lamination hull B = L(A) is open and bounded, then, for any bounded domain R n and any 2 B = L(A), there exists a map
9 AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL 9 u = u 2 W q) ( R m ) satisfying (3:) and, therefore, B = L(A) p (K) for all p<q: Proof. Condition (3.7) implies that, for any p<q, (3.8) c p =sup 2A jbj B jdv (x)j p dx Cp q q ; p < : From this, using Theorem 3.4 quoted above and its remarks, we obtain, for each given 2 B = L(A), a map u = u depending only on the power q and the family fv j 2 Ag in (3.7) that belongs to W p ( R n ) for all p<qand satises (3.) and jj jduj p dx maxfc p sup jj p g: 2B Multiplying this estimate by q ; p, taking the =p power, and maximizing over p<qshow that [Du] q maxfc q q sup jjg < 2B where q = sup pq (q ; p) =p < is a constant. This shows u = u 2 W q) ( R n ) satisfying (3.) and completes the proof. 4. Proof of Theorem.4 In this nal section, we apply Theorem 3.5 to prove the main attainment theorem, Theorem.4. To do so, as in Yan [3, 4], we further introduce certain subsets in M nn : Let W L be the set dened by (3.2). Givenanumber >0 dene (4.) (4.2) (4.3) R(n) =f 2 M nn jjj n = j det jg A = f 2 R(n) jjj <g A ~ = A nf0g B = f 2 M nn jjj <g: Proposition 4.. L n ( ~A )=B : Therefore B = L( ~A ) is a bounded open set. Proof. Let 2 B : If 6= 0 then the proof of Proposition 4. in [4] has essentially shown that 2 L n; ( ~A ): Therefore one has only to show 0 2L n ( ~A ): To prove this, let 0 <t<be xed. Note that, since the diagonal matrices ( 2 n ) and (; 2 n ), where i =
10 0 BAISHENG YAN t, are in ~A and their dierence is rank-one, by Denition 3., we have (0 2 n )= ( 2 n )+(; 2 n ) 2 belongs to L ( ~A ) for all i = t: Now that matrices (0 2 n ) and (0 ; 2 n ) are in L ( ~A )anddier by rank-one, hence (0 0 3 n )= (0 2 3 n )+(0 ; 2 3 n ) 2 is in L 2 ( ~A ) for all i = t: Repeating this argument nitely many steps, we eventually arrive at 0 = (0 0 0) 2L n ( ~A ): Proposition 4.2. Let > 0, n 3, L and q = nl. Then, L+ for each 2 ~A, there exists a map v = v 2 W q) (B R n ) satisfying Dv(x) 2 W L a.e. x 2 B and = x such that (4.4) c 0 sup 2 ~ A [Dv ] q B C(n L) <: Proof. Let 2 ~A be given. We look for map v = v in the form of radial maps: v(x) = jxj x where is a number to be chosen later. Note that all such maps satisfy = x in the W p -sense if v 2 W p (B R n ) for some p : Let r = jxj,! = r ; x then a simple calculation shows that (4.5) (4.6) (4.7) Dv(x) =r (I +!!) det Dv(x) = ( + )r n det ( r jj j+j jdv(x)j = j+jr jj j+j > : If det > 0, we choose = to be one of the two numbers dened above, and then one easily sees that Dv(x) 2 S for all x 6= 0 in B and, a computation also shows that this function v = v belongs to W q) (B R n )andsatises (4.8) [Dv ] q B C (n L)jj: If det < 0 we let = 2 = ;L ; ; and in this case one easily sees that Dv (x) 2 JS 2 for all x 6= 0: Furthermore, we compute to get jbj B jdv (x)j p dx = jj p r 2p+n; dr = jjp 0 2 p + n = L jj p L +q ; p
11 AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL which shows that v = v belongs to W q) (B R n ) and satises (4.9) [Dv ] q B C 2 (n L)jj: Combining (4.8), (4.9), one proves (4.4). The proofiscompleted. Proof of Theorem.4. Let 2 M nn be given. Let > 0 be a number such that jj < : Then 2 B : From Proposition 4., L( ~A ) = B is open and bounded. Also, from Proposition 4.2, the set A = ~A satises the condition (3.7) in Theorem 3.5 with K = W L and q = nl : Therefore, Theorem 3.5 implies that, there exists a map L+ u = u 2 W q) ( R n ) satisfying Du(x) 2 W L for a.e. x 2 and = x: Note also that [Du] q C 0 <: We complete the proof. References [] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal., 86 (984), 25{45. [2] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63 (977), 337{403. [3] J. M. Ball and R. D. James, Proposed experimental tests of a theory of ne microstructures and the two well problem, Phil. Trans. Roy. Soc. London, 338A (992), 389{450. [4] J. M. Ball and F. Murat, W p -Quasiconvexity and variational problems for multiple integrals, J. Funct. Anal., 58 (984), 225{253. [5] B. Dacorogna, \Direct Methods in the Calculus of Variations." Springer- Verlag Berlin, 989. [6] T. Iwaniec, An approach to Cauchy-Riemann operators in R n, preprint. [7] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal., 9 (992), 29{43. [8] C. B. Morrey, \Multiple Integrals in the Calculus of Variations." Springer- Verlag Berlin, 966. [9] S. Muller, Variational models for microstructure and phase transitions, in \Calculus of Variations and Geometric Evolution Problems," (Cetraro, 996), pp. 85{20, Lecture Notes in Math. 73, Springer, 999. [0] S. Muller and V. Sverak, Attainment results for the two-well problem by convex integration, Geometric analysis and the calculus of variations, 239{ 25, Internat. Press Cambridge, MA, 996. [] V. Sverak, On the problem of two wells, in \Microstructure and Phase Transition," pp. 83{90, (D. Kinderlehrer et al eds.), Springer-Verlag New York, 993. [2] B. Yan, On rank-one convex and polyconvex conformal energy functions with slow growth, Proc. Roy. Soc. Edinburgh, 27A (997), 65{663. [3] B. Yan, Linear boundary values of weakly quasiregular mappings, C. R. Acad. Sci. Paris Ser. I Math., 33 (2000), 379{384.
12 2 BAISHENG YAN [4] B. Yan, A linear boundary value problem for weakly quasiregular mappings, Cal. Var. Partial Dierential Equations, to appear. [5] B. Yan, Semiconvex hulls of quasiconformal sets, J. Convex Analysis, to appear. [6] B. Yan and. hou, Stability of weakly almost conformal mappings, Proc. Amer. Math. Soc., 26 (998), 48{489. [7] B. Yan and. hou, L p -mean coercivity, regularity and relaxation in the calculus of variations, Nonlinear Analysis, to appear. [8] K. hang, On various semiconvex hulls in the calculus of variations, Cal. Var. Partial Dierential Equations, 6 (998), 43{60. [9] K. hang, Rank-one connections at innity and quasiconvex hulls, J. Convex Analysis, 7 (2000), 9{45. Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA address: yan@math.msu.edu
ON TRIVIAL GRADIENT YOUNG MEASURES BAISHENG YAN Abstract. We give a condition on a closed set K of real nm matrices which ensures that any W 1 p -grad
ON TRIVIAL GRAIENT YOUNG MEASURES BAISHENG YAN Abstract. We give a condition on a closed set K of real nm matrices which ensures that any W 1 p -gradient Young measure supported on K must be trivial the
More information2 BAISHENG YAN The gradient matrix Du(x) is a well-dened n m-matrix for almost every x in : Therefore, in the sequel, the pointwise relations like (1.
RELAXATION OF CERTAIN PARTIAL DIFFERENTIAL RELATIONS IN SOBOLEV SPACES BAISHENG YAN Abstract. Many mappings in geometry, mechanics and control theory satisfy certain relations among their derivatives.
More informationEstimates of Quasiconvex Polytopes in the Calculus of Variations
Journal of Convex Analysis Volume 13 26), No. 1, 37 5 Estimates of Quasiconvex Polytopes in the Calculus of Variations Kewei Zhang epartment of Mathematics, University of Sussex, Falmer, Brighton, BN1
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 informationRegularity of Minimizers in Nonlinear Elasticity the Case of a One-Well Problem in Nonlinear Elasticity
TECHNISCHE MECHANIK, 32, 2-5, (2012), 189 194 submitted: November 1, 2011 Regularity of Minimizers in Nonlinear Elasticity the Case of a One-Well Problem in Nonlinear Elasticity G. Dolzmann In this note
More informationON SELECTION OF SOLUTIONS TO VECTORIAL HAMILTON-JACOBI SYSTEM
ON SELECTION OF SOLUTIONS TO VECTORIAL HAMILTON-JACOBI SYSTEM BAISHENG YAN Abstract. In this paper, we discuss some principles on the selection of the solutions of a special vectorial Hamilton-Jacobi system
More informationAn O(n) invariant rank 1 convex function that is not polyconvex
THEORETICAL AND APPLIED MECHANICS vol. 28-29, pp. 325-336, Belgrade 2002 Issues dedicated to memory of the Professor Rastko Stojanović An O(n) invariant rank 1 convex function that is not polyconvex M.
More informationLusin type theorem with quasiconvex hulls
Lusin type theorem with quasiconvex hulls Agnieszka Ka lamajska Institute of Mathematics, Warsaw University, ul. Banacha 2, 02 097 Warszawa, Poland We obtain Lusin type theorem showing that after extracting
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 informationTwo-dimensional examples of rank-one convex functions that are not quasiconvex
ANNALES POLONICI MATHEMATICI LXXIII.3(2) Two-dimensional examples of rank-one convex functions that are not quasiconvex bym.k.benaouda (Lille) andj.j.telega (Warszawa) Abstract. The aim of this note is
More informationANNALES DE L I. H. P., SECTION C
ANNALES DE L I. H. P., SECTION C BAISHENG YAN Remarks on W 1,p -stability of the conformal set in higher dimensions Annales de l I. H. P., section C, tome 13, n o 6 (1996), p. 691-705.
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 informationA NUMERICAL ITERATIVE SCHEME FOR COMPUTING FINITE ORDER RANK-ONE CONVEX ENVELOPES
A NUMERICAL ITERATIVE SCHEME FOR COMPUTING FINITE ORDER RANK-ONE CONVEX ENVELOPES XIN WANG, ZHIPING LI LMAM & SCHOOL OF MATHEMATICAL SCIENCES, PEKING UNIVERSITY, BEIJING 87, P.R.CHINA Abstract. It is known
More informationZHIPING LI. from each other under certain general hypotheses such as uniform lower
ESAIM: Control, Optimisation Calculus of Variations UL: http://www.emath.fr/cocv/ June 1996, vol. 1, pp. 169-189 LOWE SEMICONTINUITY OF MULTIPLE INTEGALS AND CONVEGENT INTEGANDS HIPING LI Abstract. Lower
More informationMath 497 R1 Winter 2018 Navier-Stokes Regularity
Math 497 R Winter 208 Navier-Stokes Regularity Lecture : Sobolev Spaces and Newtonian Potentials Xinwei Yu Jan. 0, 208 Based on..2 of []. Some ne properties of Sobolev spaces, and basics of Newtonian potentials.
More informationTartar s conjecture and localization of the quasiconvex hull in R 2 2
Tartar s conjecture and localization of the quasiconvex hull in R 2 2 Daniel Faraco and László Székelyhidi Jr. Abstract We give a concrete and surprisingly simple characterization of compact sets K R 2
More informationSome nonlinear elliptic equations in R N
Nonlinear Analysis 39 000) 837 860 www.elsevier.nl/locate/na Some nonlinear elliptic equations in Monica Musso, Donato Passaseo Dipartimento di Matematica, Universita di Pisa, Via Buonarroti,, 5617 Pisa,
More informationMax-Planck-Institut fur Mathematik in den Naturwissenschaften Leipzig A sharp version of hang's theorem on truncating sequences of gradients by Stefan Muller Preprint-Nr.: 18 1997 A sharp version of hang's
More informationHOMEOMORPHISMS OF BOUNDED VARIATION
HOMEOMORPHISMS OF BOUNDED VARIATION STANISLAV HENCL, PEKKA KOSKELA AND JANI ONNINEN Abstract. We show that the inverse of a planar homeomorphism of bounded variation is also of bounded variation. In higher
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 informationBifurcation from the rst eigenvalue of some nonlinear elliptic operators in Banach spaces
Nonlinear Analysis 42 (2000) 561 572 www.elsevier.nl/locate/na Bifurcation from the rst eigenvalue of some nonlinear elliptic operators in Banach spaces Pavel Drabek a;, Nikos M. Stavrakakis b a Department
More informationRearrangements and polar factorisation of countably degenerate functions G.R. Burton, School of Mathematical Sciences, University of Bath, Claverton D
Rearrangements and polar factorisation of countably degenerate functions G.R. Burton, School of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, U.K. R.J. Douglas, Isaac Newton
More informationBoundedly complete weak-cauchy basic sequences in Banach spaces with the PCP
Journal of Functional Analysis 253 (2007) 772 781 www.elsevier.com/locate/jfa Note Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP Haskell Rosenthal Department of Mathematics,
More informationA note on the isoperimetric inequality
A note on the isoperimetric inequality Jani Onninen Abstract n 2 We show that the sharp integral form on the isoperimetric inequality holds for those orientation preserving mappings f W (Ω, R n ) whose
More informationSCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze
ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze KEWEI ZHANG A construction of quasiconvex functions with linear growth at infinity Annali della Scuola Normale Superiore di Pisa, Classe
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 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 informationSOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES
ARCHIVUM MATHEMATICUM (BRNO) Tomus 42 (2006), 167 174 SOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES ABDELHAKIM MAADEN AND ABDELKADER STOUTI Abstract. It is shown that under natural assumptions,
More informationGarrett: `Bernstein's analytic continuation of complex powers' 2 Let f be a polynomial in x 1 ; : : : ; x n with real coecients. For complex s, let f
1 Bernstein's analytic continuation of complex powers c1995, Paul Garrett, garrettmath.umn.edu version January 27, 1998 Analytic continuation of distributions Statement of the theorems on analytic continuation
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 information/00 $ $.25 per page
Contemporary Mathematics Volume 00, 0000 Domain Decomposition For Linear And Nonlinear Elliptic Problems Via Function Or Space Decomposition UE-CHENG TAI Abstract. In this article, we use a function decomposition
More informationWeak Convergence Methods for Energy Minimization
Weak Convergence Methods for Energy Minimization Bo Li Department of Mathematics University of California, San Diego E-mail: bli@math.ucsd.edu June 3, 2007 Introduction This compact set of notes present
More informationCOMPACT DIFFERENCE OF WEIGHTED COMPOSITION OPERATORS ON N p -SPACES IN THE BALL
COMPACT DIFFERENCE OF WEIGHTED COMPOSITION OPERATORS ON N p -SPACES IN THE BALL HU BINGYANG and LE HAI KHOI Communicated by Mihai Putinar We obtain necessary and sucient conditions for the compactness
More informationON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT
PORTUGALIAE MATHEMATICA Vol. 56 Fasc. 3 1999 ON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT M. Guedda Abstract: In this paper we consider the problem u = λ u u + f in, u = u
More informationu( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)
M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation
More informationLARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS. S. G. Bobkov and F. L. Nazarov. September 25, 2011
LARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS S. G. Bobkov and F. L. Nazarov September 25, 20 Abstract We study large deviations of linear functionals on an isotropic
More informationLocal invertibility in Sobolev spaces. Carlos Mora-Corral
1/24 Local invertibility in Sobolev spaces Carlos Mora-Corral University Autonoma of Madrid (joint work with Marco Barchiesi and Duvan Henao) 2/24 Nonlinear Elasticity Calculus of Variations approach A
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More informationChain Rule for planar bi Sobolev maps
Chain Rule for planar bi Sobolev maps Nota di L. D Onofrio, L. Migliaccio, C. Sbordone and R. Schiattarella Presentata dal socio Carlo Sbordone (Adunanza del 7 febbraio 2014) Abstract - For a planar bi
More informationL F(P + Drf>(x))dx ~ F(P)meas(G),
259 MULTIWELL PROBLEMS AND RESTRICTIONS ON MICROSTRUCTURE KEWEI ZHANG 1. Problems and Background. Consider a variational problem: (1). min f F(Du(x))dx, u=u0 on8fl } 0 and suppose the integrand F is continuous
More informationANNALES DE L I. H. P., SECTION C
ANNALES DE L I. H. P., SECTION C JAN KRISTENSEN On the non-locality of quasiconvexity Annales de l I. H. P., section C, tome 16, n o 1 (1999), p. 1-13
More informationATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS. Emerson A. M. de Abreu Alexandre N.
ATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS Emerson A. M. de Abreu Alexandre N. Carvalho Abstract Under fairly general conditions one can prove that
More informationThe Erwin Schrodinger International Pasteurgasse 6/7. Institute for Mathematical Physics A-1090 Wien, Austria
ESI The Erwin Schrodinger International Pasteurgasse 6/7 Institute for Mathematical Physics A-1090 Wien, Austria On the Point Spectrum of Dirence Schrodinger Operators Vladimir Buslaev Alexander Fedotov
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 informationOn a weighted total variation minimization problem
On a weighted total variation minimization problem Guillaume Carlier CEREMADE Université Paris Dauphine carlier@ceremade.dauphine.fr Myriam Comte Laboratoire Jacques-Louis Lions, Université Pierre et Marie
More informationSemi-strongly asymptotically non-expansive mappings and their applications on xed point theory
Hacettepe Journal of Mathematics and Statistics Volume 46 (4) (2017), 613 620 Semi-strongly asymptotically non-expansive mappings and their applications on xed point theory Chris Lennard and Veysel Nezir
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 informationEXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM
EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM JENICĂ CRÎNGANU We derive existence results for operator equations having the form J ϕu = N f u, by using
More informationMax-Planck-Institut fur Mathematik in den Naturwissenschaften Leipzig Conditions for equality of hulls in the calculus of variations by Georg Dolzmann, Bernd Kirchheim and Jan Kristensen Preprint no.:
More informationA PROPERTY OF SOBOLEV SPACES ON COMPLETE RIEMANNIAN MANIFOLDS
Electronic Journal of Differential Equations, Vol. 2005(2005), No.??, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A PROPERTY
More informationFrom Isometric Embeddings to Turbulence
From Isometric Embeddings to Turbulence László Székelyhidi Jr. (Bonn) Programme for the Cours Poupaud 15-17 March 2010, Nice Monday Morning Lecture 1. The Nash-Kuiper Theorem In 1954 J.Nash shocked the
More informationLocal mountain-pass for a class of elliptic problems in R N involving critical growth
Nonlinear Analysis 46 (2001) 495 510 www.elsevier.com/locate/na Local mountain-pass for a class of elliptic problems in involving critical growth C.O. Alves a,joão Marcos do O b; ;1, M.A.S. Souto a;1 a
More informationSpurious Chaotic Solutions of Dierential. Equations. Sigitas Keras. September Department of Applied Mathematics and Theoretical Physics
UNIVERSITY OF CAMBRIDGE Numerical Analysis Reports Spurious Chaotic Solutions of Dierential Equations Sigitas Keras DAMTP 994/NA6 September 994 Department of Applied Mathematics and Theoretical Physics
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 informationHIGHER INTEGRABILITY WITH WEIGHTS
Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 355 366 HIGHER INTEGRABILITY WITH WEIGHTS Juha Kinnunen University of Jyväskylä, Department of Mathematics P.O. Box 35, SF-4351
More informationON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS. G. Makay
ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS G. Makay Student in Mathematics, University of Szeged, Szeged, H-6726, Hungary Key words and phrases: Lyapunov
More informationCOINCIDENCE SETS IN THE OBSTACLE PROBLEM FOR THE p-harmonic OPERATOR
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 95, Number 3, November 1985 COINCIDENCE SETS IN THE OBSTACLE PROBLEM FOR THE p-harmonic OPERATOR SHIGERU SAKAGUCHI Abstract. We consider the obstacle
More informationFunctions: A Fourier Approach. Universitat Rostock. Germany. Dedicated to Prof. L. Berg on the occasion of his 65th birthday.
Approximation Properties of Multi{Scaling Functions: A Fourier Approach Gerlind Plona Fachbereich Mathemati Universitat Rostoc 1851 Rostoc Germany Dedicated to Prof. L. Berg on the occasion of his 65th
More informationSOME MEASURABILITY AND CONTINUITY PROPERTIES OF ARBITRARY REAL FUNCTIONS
LE MATEMATICHE Vol. LVII (2002) Fasc. I, pp. 6382 SOME MEASURABILITY AND CONTINUITY PROPERTIES OF ARBITRARY REAL FUNCTIONS VITTORINO PATA - ALFONSO VILLANI Given an arbitrary real function f, the set D
More informationThuong Nguyen. SADCO Internal Review Metting
Asymptotic behavior of singularly perturbed control system: non-periodic setting Thuong Nguyen (Joint work with A. Siconolfi) SADCO Internal Review Metting Rome, Nov 10-12, 2014 Thuong Nguyen (Roma Sapienza)
More informationA compactness theorem for Yamabe metrics
A compactness theorem for Yamabe metrics Heather acbeth November 6, 2012 A well-known corollary of Aubin s work on the Yamabe problem [Aub76a] is the fact that, in a conformal class other than the conformal
More informationCOMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS
Dynamic Systems and Applications 22 (203) 37-384 COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS VICENŢIU D. RĂDULESCU Simion Stoilow Mathematics Institute
More informationQUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE. Denitizable operators in Krein spaces have spectral properties similar to those
QUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE BRANKO CURGUS and BRANKO NAJMAN Denitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces.
More informationMonotonicity formulas for variational problems
Monotonicity formulas for variational problems Lawrence C. Evans Department of Mathematics niversity of California, Berkeley Introduction. Monotonicity and entropy methods. This expository paper is a revision
More informationb i (µ, x, s) ei ϕ(x) µ s (dx) ds (2) i=1
NONLINEAR EVOLTION EQATIONS FOR MEASRES ON INFINITE DIMENSIONAL SPACES V.I. Bogachev 1, G. Da Prato 2, M. Röckner 3, S.V. Shaposhnikov 1 The goal of this work is to prove the existence of a solution to
More informationWeak solutions for some quasilinear elliptic equations by the sub-supersolution method
Nonlinear Analysis 4 (000) 995 100 www.elsevier.nl/locate/na Weak solutions for some quasilinear elliptic equations by the sub-supersolution method Manuel Delgado, Antonio Suarez Departamento de Ecuaciones
More informationA Caffarelli-Kohn-Nirenberg type inequality with variable exponent and applications to PDE s
A Caffarelli-Kohn-Nirenberg type ineuality with variable exponent and applications to PDE s Mihai Mihăilescu a,b Vicenţiu Rădulescu a,c Denisa Stancu-Dumitru a a Department of Mathematics, University of
More informationOn It^o's formula for multidimensional Brownian motion
On It^o's formula for multidimensional Brownian motion by Hans Follmer and Philip Protter Institut fur Mathemati Humboldt-Universitat D-99 Berlin Mathematics and Statistics Department Purdue University
More informationON MAPS WHOSE DISTRIBUTIONAL JACOBIAN IS A MEASURE
[version: December 15, 2007] Real Analysis Exchange Summer Symposium 2007, pp. 153-162 Giovanni Alberti, Dipartimento di Matematica, Università di Pisa, largo Pontecorvo 5, 56127 Pisa, Italy. Email address:
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 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 informationConvergence rate estimates for the gradient differential inclusion
Convergence rate estimates for the gradient differential inclusion Osman Güler November 23 Abstract Let f : H R { } be a proper, lower semi continuous, convex function in a Hilbert space H. The gradient
More informationGinés López 1, Miguel Martín 1 2, and Javier Merí 1
NUMERICAL INDEX OF BANACH SPACES OF WEAKLY OR WEAKLY-STAR CONTINUOUS FUNCTIONS Ginés López 1, Miguel Martín 1 2, and Javier Merí 1 Departamento de Análisis Matemático Facultad de Ciencias Universidad de
More informationGlobal unbounded solutions of the Fujita equation in the intermediate range
Global unbounded solutions of the Fujita equation in the intermediate range Peter Poláčik School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Eiji Yanagida Department of Mathematics,
More informationSobolev Spaces. Chapter 10
Chapter 1 Sobolev Spaces We now define spaces H 1,p (R n ), known as Sobolev spaces. For u to belong to H 1,p (R n ), we require that u L p (R n ) and that u have weak derivatives of first order in L p
More informationMax-Planck-Institut fur Mathematik in den Naturwissenschaften Leipzig A nonlocal anisotropic model for phase transitions Part II: Asymptotic behaviour of rescaled energies by Giovanni Alberti and Giovanni
More information1 Which sets have volume 0?
Math 540 Spring 0 Notes #0 More on integration Which sets have volume 0? The theorem at the end of the last section makes this an important question. (Measure theory would supersede it, however.) Theorem
More informationYuqing Chen, Yeol Je Cho, and Li Yang
Bull. Korean Math. Soc. 39 (2002), No. 4, pp. 535 541 NOTE ON THE RESULTS WITH LOWER SEMI-CONTINUITY Yuqing Chen, Yeol Je Cho, and Li Yang Abstract. In this paper, we introduce the concept of lower semicontinuous
More informationNonlinear elliptic systems with exponential nonlinearities
22-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 9, 22, pp 139 147. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationThis method is introduced by the author in [4] in the case of the single obstacle problem (zero-obstacle). In that case it is enough to consider the v
Remarks on W 2;p -Solutions of Bilateral Obstacle Problems Srdjan Stojanovic Department of Mathematical Sciences University of Cincinnati Cincinnati, OH 4522-0025 November 30, 995 Abstract The well known
More informationA Note on the Class of Superreflexive Almost Transitive Banach Spaces
E extracta mathematicae Vol. 23, Núm. 1, 1 6 (2008) A Note on the Class of Superreflexive Almost Transitive Banach Spaces Jarno Talponen University of Helsinki, Department of Mathematics and Statistics,
More informationTHE LARGEST INTERSECTION LATTICE OF A CHRISTOS A. ATHANASIADIS. Abstract. We prove a conjecture of Bayer and Brandt [J. Alg. Combin.
THE LARGEST INTERSECTION LATTICE OF A DISCRIMINANTAL ARRANGEMENT CHRISTOS A. ATHANASIADIS Abstract. We prove a conjecture of Bayer and Brandt [J. Alg. Combin. 6 (1997), 229{246] about the \largest" intersection
More informationPartial Differential Equations, 2nd Edition, L.C.Evans The Calculus of Variations
Partial Differential Equations, 2nd Edition, L.C.Evans Chapter 8 The Calculus of Variations Yung-Hsiang Huang 2018.03.25 Notation: denotes a bounded smooth, open subset of R n. All given functions are
More informationOn the role playedby the Fuck spectrum in the determination of critical groups in elliptic problems where the asymptotic limits may not exist
Nonlinear Analysis 49 (2002) 603 611 www.elsevier.com/locate/na On the role playedby the Fuck spectrum in the determination of critical groups in elliptic problems where the asymptotic limits may not exist
More informationMATH 426, TOPOLOGY. p 1.
MATH 426, TOPOLOGY THE p-norms In this document we assume an extended real line, where is an element greater than all real numbers; the interval notation [1, ] will be used to mean [1, ) { }. 1. THE p
More information2 G. D. DASKALOPOULOS AND R. A. WENTWORTH general, is not true. Thus, unlike the case of divisors, there are situations where k?1 0 and W k?1 = ;. r;d
ON THE BRILL-NOETHER PROBLEM FOR VECTOR BUNDLES GEORGIOS D. DASKALOPOULOS AND RICHARD A. WENTWORTH Abstract. On an arbitrary compact Riemann surface, necessary and sucient conditions are found for the
More informationMultiple positive solutions for a class of quasilinear elliptic boundary-value problems
Electronic Journal of Differential Equations, Vol. 20032003), No. 07, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp) Multiple positive
More informationContractive metrics for scalar conservation laws
Contractive metrics for scalar conservation laws François Bolley 1, Yann Brenier 2, Grégoire Loeper 34 Abstract We consider nondecreasing entropy solutions to 1-d scalar conservation laws and show that
More informationExistence of a ground state and blow-up problem for a nonlinear Schrödinger equation with critical growth
Existence of a ground state and blow-up problem for a nonlinear Schrödinger equation with critical growth Takafumi Akahori, Slim Ibrahim, Hiroaki Kikuchi and Hayato Nawa 1 Introduction In this paper, we
More informationSTOCHASTIC DIFFERENTIAL EQUATIONS WITH EXTRA PROPERTIES H. JEROME KEISLER. Department of Mathematics. University of Wisconsin.
STOCHASTIC DIFFERENTIAL EQUATIONS WITH EXTRA PROPERTIES H. JEROME KEISLER Department of Mathematics University of Wisconsin Madison WI 5376 keisler@math.wisc.edu 1. Introduction The Loeb measure construction
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-1090 Wien, Austria Common Homoclinic Points of Commuting Toral Automorphisms Anthony Manning Klaus Schmidt
More informationMax-Planck-Institut fur Mathematik in den Naturwissenschaften Leipzig Uniformly distributed measures in Euclidean spaces by Bernd Kirchheim and David Preiss Preprint-Nr.: 37 1998 Uniformly Distributed
More informationEXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 05, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF NONTRIVIAL
More informationSimple Lie subalgebras of locally nite associative algebras
Simple Lie subalgebras of locally nite associative algebras Y.A. Bahturin Department of Mathematics and Statistics Memorial University of Newfoundland St. John's, NL, A1C5S7, Canada A.A. Baranov Department
More informationSYMMETRY IN REARRANGEMENT OPTIMIZATION PROBLEMS
Electronic Journal of Differential Equations, Vol. 2009(2009), No. 149, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SYMMETRY IN REARRANGEMENT
More informationA nodal solution of the scalar field equation at the second minimax level
Bull. London Math. Soc. 46 (2014) 1218 1225 C 2014 London Mathematical Society doi:10.1112/blms/bdu075 A nodal solution of the scalar field equation at the second minimax level Kanishka Perera and Cyril
More informationThe Pacic Institute for the Mathematical Sciences http://www.pims.math.ca pims@pims.math.ca Surprise Maximization D. Borwein Department of Mathematics University of Western Ontario London, Ontario, Canada
More informationOn an algebra related to orbit-counting. Peter J. Cameron. Queen Mary and Westeld College. London E1 4NS U.K. Abstract
On an algebra related to orbit-counting Peter J. Cameron School of Mathematical Sciences Queen Mary and Westeld College London E1 4NS U.K. Abstract With any permutation group G on an innite set is associated
More informationHopfLax-Type Formula for u t +H(u, Du)=0*
journal of differential equations 126, 4861 (1996) article no. 0043 HopfLax-Type Formula for u t +H(u, Du)=0* E. N. Barron, - R. Jensen, and W. Liu 9 Department of Mathematical Sciences, Loyola University,
More informationStanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon Measures
2 1 Borel Regular Measures We now state and prove an important regularity property of Borel regular outer measures: Stanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon
More information