A PROPERTY OF SOBOLEV SPACES ON COMPLETE RIEMANNIAN MANIFOLDS
|
|
- Dustin Elliott
- 5 years ago
- Views:
Transcription
1 Electronic Journal of Differential Equations, Vol. 2005(2005), No.??, pp ISSN: URL: or ftp ejde.math.txstate.edu (login: ftp) A PROPERTY OF SOBOLEV SPACES ON COPLETE RIEANNIAN ANIFOLDS OGNJEN ILATOVIC Abstract. Let (, g) be a complete Riemannian manifold with metric g and the Riemannian volume form dν. We consider the R k -valued functions T [W 1,2 () L 1 loc ()]k and u [W 1,2 ()] k on, where [W 1,2 ()] k is a Sobolev space on and [W 1,2 ()] k is its dual. We give a sufficient condition for the equality of T, u and the integral of (T u) over, where, is the duality between [W 1,2 ()] k and [W 1,2 ()] k. This is an extension to complete Riemannian manifolds of a result of H. Brézis and F. E. Browder. 1. Introduction and main result The setting. Let (, g) be a C Riemannian manifold without boundary, with metric g = (g jk ) and dim = n. We will assume that is connected, oriented, and complete. By dν we will denote the Riemannian volume element of. In any local coordinates x 1,..., x n, we have dν = det(g jk ) dx 1 dx 2... dx n. By L 2 () we denote the space of real-valued square integrable functions on with the inner product (u, v) = (uv) dν. Unless specified otherwise, in all function spaces below, the functions are realvalued. In what follows, C () denotes the space of smooth functions on, Cc () denotes the space of smooth compactly supported functions on, Ω 1 () denotes the space of smooth 1-forms on, and L 2 (Λ 1 T ) denotes the space of square integrable 1-forms on. By W 1,2 () we denote the completion of Cc () in the norm u 2 W = u 2 dν + du 2 dν, 1,2 where d : C () Ω 1 () is the standard differential. Remark 1.1. It is well known (see, for example, Chapter 2 in [1]) that if (, g) is a complete Riemannian manifold, then W 1,2 () = {u L 2 (): du L 2 (Λ 1 T )} athematics Subject Classification. 58J05. Key words and phrases. Complete Riemannian manifold; Sobolev space. c 2005 Texas State University - San arcos. Submitted June 25, Published??. 1
2 2 O. ILATOVIC EJDE-2005/?? By W 1,2 () we denote the dual space of W 1,2 (), and by, we will denote the duality between W 1,2 () and W 1,2 (). In what follows, [Cc ()] k, [L 2 ()] k, [L 2 (Λ 1 T )] k and [W 1,2 ()] k denote the space of all ordered k-tuples u = (u 1, u 2,..., u k ) such that u j Cc (), u j L 2 (), u j L 2 (Λ 1 T ), u j W 1,2 (), respectively, for all 1 j k. For u [W 1,2 ()] k, we will use the following notation: du := (du 1, du 2,..., du k ), (1.1) u := (u u u 2 k) 1/2, (1.2) du := ( du du du k 2 ) 1/2, (1.3) where du j denotes the length of the cotangent vector du j. The space [W 1,2 ()] k is the completion of [Cc ()] k in the norm u 2 [W 1,2 ()] = u 2 dν + du 2 dν, k where u and du are as in (1.2) and (1.3) respectively. Remark 1.2. As in Remark 1.1, if (, g) is a complete Riemannian manifold, then [W 1,2 ()] k = {u [L 2 ()] k : du [L 2 (Λ 1 T )] k }. Assumption (H1). Assume that (1) u = (u 1, u 2,..., u k ) [W 1,2 ()] k and (2) T = (T 1, T 2,..., T k ), where T 1, T 2,..., T k W 1,2 () L 1 loc (). Here, the notation T j W 1,2 () L 1 loc () means that T j is a.e. defined function belonging to L 1 loc () such that φ T j φ dν, φ Cc (), extends continuously to W 1,2 (). For a.e. x, denote (T u)(x) := T, u := T j (x)u j (x), (1.4) T j, u j, (1.5) where, on the right hand side of (1.5) denotes the duality between W 1,2 () and W 1,2 (). We now state our main result. Theorem 1.3. Assume that (, g) is a complete Riemannian manifold. Assume that u = (u 1, u 2,..., u k ) and T = (T 1, T 2,..., T k ) satisfy the assumption (H1). Assume that there exists a function f L 1 () such that Then (T u) L 1 () and T, u = (T u)(x) f(x), a.e. on. (1.6) (T u)(x) dν(x).
3 EJDE-2005/?? A PROPERTY OF SOBOLEV SPACES 3 In the following Corollary, by W 1,2 (, C), W 1,2 (, C) and L 1 loc (, C) we denote the complex analogues of spaces W 1,2 (), W 1,2 () and L 1 loc (). By, we denote the Hermitian duality between W 1,2 (, C) and W 1,2 (, C). Corollary 1.4. Assume that (, g) is a complete Riemannian manifold. Assume that T W 1,2 (, C) L 1 loc (, C) and u W 1,2 (, C). Assume that there exists a real-valued function f L 1 () such that Re(T ū) f, a.e. on. Then Re(T ū) L 1 () and Re T, u = Re(T ū) dν. Remark 1.5. Theorem 1.3 and Corollary 1.4 extend the corresponding results of H. Brézis and F. E. Browder [3] from R n to complete Riemannian manifolds. The results of [3] were used, among other applications, in studying self-adjointness and m-accretivity in L 2 (R n, C) of Schrödinger operators with singular potentials; see, for example, H. Brézis and T. Kato [4]. Analogously, Theorem 1.3 and Corollary 1.4 can be used in the study of self-adjoint and m-accretive realizations (in the space L 2 (, C)) of Schrödinger-type operators with singular potentials, where is a complete Riemannian manifold, as well as in the study of partial differential equations on complete Riemannian manifolds. 2. Proof of Theorem 1.3 We will adopt the arguments of H. Brézis and F. E. Browder [3] to the context of a complete Riemannian manifold. In what follows, F : R k R l is a C 1 vectorvalued function F (y) = (F 1 (y), F 2 (y),..., F l (y)). By df (y) we will denote the derivative of F at y = (y 1, y 2,..., y k ). Lemma 2.1. Assume that F C 1 (R k, R l ), F (0) = 0, and for all y R k, df (y) C where C 0 is a constant. Assume that u = (u 1, u 2,..., u k ) [W 1,2 ()] k. Then (F u) [W 1,2 ()] l, and the following holds: where d(f u) = F u j du j, (2.1) F ( F1 = (u), F 2 (u),..., F ) l (u). (2.2) u j (Here the notation F s (u), where 1 s l, denotes the composition of F s and u. The notation d(f u) denotes the ordered l-tuple (d(f 1 u), d(f 2 u),..., d(f l u)), where d(f s u), 1 s l, is the differential of the scalar-valued function F s u on. Proof. Let u [W 1,2 ()] k. By definition of [W 1,2 ()] k, the weak derivatives du j, 1 j k, exist and du j L 2 (). By Lemma 7.5 in [6], it follows that for all
4 4 O. ILATOVIC EJDE-2005/?? 1 s l, the following holds: where d(f s u) = F s u j F s u j du j, = F s (u). This shows (2.1). Since df is bounded and since du j L 2 (Λ 1 T ), it follows that d(f s u) L 2 (Λ 1 T ) for all 1 s l. Thus d(f u) [L 2 (Λ 1 T )] l. oreover, since u [W 1,2 ()] k and F s u = F s (u) F s (0) C 1 u, where C 1 0 is a constant and u is as in (1.2), it follows that (F s u) L 2 () for all 1 s l. Thus (F u) [L 2 ()] l. Therefore, (F u) [W 1,2 ()] l, and the Lemma is proven. Lemma 2.2. Assume that u, v W 1,2 () L (). Then (uv) W 1,2 () and d(uv) = (du)v + u(dv). (2.3) Proof. By the remark after the equation (7.18) in [6], the equation (2.3) holds if the weak derivatives du, dv exist and if uv L 1 loc () and ((du)v + u(dv)) L1 loc (). By the hypotheses of the Lemma, these conditions are satisfied, and, hence, (2.3) holds. Furthermore, since u, v W 1,2 () L (), we have (uv) L 2 (). By hypotheses of the Lemma and by (2.3) we have d(uv) L 2 (). Thus (uv) W 1,2 (), and the Lemma is proven. In the next lemma, the statement f : R R is a piecewise smooth function means that f is continuous and has piecewise continuous first derivative. Lemma 2.3. Assume that f : R R is a piecewise smooth function with f(0) = 0 and f L (R). Let S denote the set of corner points of f. Assume that u W 1,2 (). Then (f u) W 1,2 () and { f (u) du for all x such that u(x) / S d(f u) = 0 for all x such that u(x) S Proof. By the remark in the second paragraph below the equation (7.24) in [6], the Lemma follows immediately from Theorem 7.8 in [6]. The following Corollary follows immediately from Lemma 2.3. Corollary 2.4. Assume that u W 1,2 (). Then u W 1,2 () and { f (u) du for all x such that u(x) 0 d u = 0 for all x such that u(x) = 0, where f(t) = t, t R.
5 EJDE-2005/?? A PROPERTY OF SOBOLEV SPACES 5 Remark 2.5. Let f(t) = t, t R. Let c be a real number. By Lemma 7.7 in [6] and by Corollary 2.4, we can write d u = h(u)du a.e. on, where { f (t) for all t 0 h(t) = c otherwise. Lemma 2.6. Assume that u, v W 1,2 () and let Then w W 1,2 () and w(x) := min{u(x), v(x)}. dw max{ du, dv }, a.e. on, where du(x) denotes the norm of the cotangent vector du(x). Proof. We can write w(x) = 1 (u(x) + v(x) u(x) v(x) ). 2 Since u, v W 1,2 (), by Corollary 2.4 we have u v W 1,2 (), and, thus, w W 1,2 (). By Remark 2.5, we have dw(x) = 1 (du(x) + dv(x) (h(u v)) (du(x) dv(x))), 2 a.e. on, (2.4) where h is as in Remark 2.5. Considering dw(x) on sets {x: u(x) > v(x)}, {x: u(x) < v(x)} and {x: u(x) = v(x)}, and using (2.4), we get dw(x) max{ du(x), dv(x) }, a.e. on. This concludes the proof of the Lemma. Lemma 2.7. Let a > 0. Let u = (u 1, u 2,..., u k ) be in [W 1,2 ()] k, let v = (v 1, v 2,..., v k ) be in [W 1,2 () L ()] k, and let ( ) w := ( u 2 + a 2 ) 1/2 min{( u 2 + a 2 ) 1/2 a, ( v 2 + a 2 ) 1/2 a} u, where u is as in (1.2). Then w [W 1,2 () L ()] k and where du is as in (1.3). dw 3 max{ du, dv }, a.e. on, Proof. Let φ = ( u 2 + a 2 ) 1/2 u. Then φ = F u, where F : R k R k is defined by F (y) = ( y 2 + a 2 ) 1/2 y, y R k. Clearly, F C 1 (R k, R k ) and F (0) = 0. It easily checked that the component functions F s (y) = ( y 2 + a 2 ) 1/2 y s satisfy F s = { ( y 2 + a 2 ) 3/2 y s y j for s j ( y 2 + a 2 ) 3/2 ( y 2 yj 2 + a2 ) for s = j. Therefore, for all 1 s, j k, we have F s (y) 1 a,
6 6 O. ILATOVIC EJDE-2005/?? and, hence, F satisfies the hypotheses of Lemma 2.1. Thus, by Lemma 2.1 we have (F u) = φ [W 1,2 ()] k. We now write the formula for dφ = (dφ 1, dφ 2,..., dφ k ). We have dφ = ( u 2 + a 2 ) 3/2( ( ) ( u 2 + a 2 )du u j du j )u, (2.5) where du is as in (1.1). By (2.5), using triangle inequality and Cauchy-Schwarz inequality, we have dφ ( u 2 + a 2 ) 3/2( ( u 2 + a 2 ) du + ) u j du j u ( u 2 + a 2 ) 3/2 ( ( u 2 + a 2 ) du + u du u ) ( u 2 + a 2 ) 3/2 ( ( u 2 + a 2 ) du + ( u 2 + a 2 ) du ) = 2( u 2 + a 2 ) 1/2 du, a.e. on, (2.6) where du j is the norm of the cotangent vector du j, and u and du are as in (1.2) and (1.3) respectively. Let ψ := min{( u 2 + a 2 ) 1/2 a, ( v 2 + a 2 ) 1/2 a}. Then ( u 2 + a 2 ) 1/2 a = G u and ( v 2 + a 2 ) 1/2 a = G v, where G(y) = ( y 2 + a 2 ) 1/2 a, y R k. Clearly, G C 1 (R k, R) and G(0) = 0, and G = ( y 2 + a 2 ) 1/2 y j, It is easily seen that there exists a constant C 2 0 such that dg(y) C 2 for all y R k. Hence, by Lemma 2.1 we have (G u) W 1,2 () and (G v) W 1,2 (). Thus, by Lemma 2.6 we have ψ W 1,2 (), and dψ max { d(( u 2 + a 2 ) 1/2 a), d(( v 2 + a 2 ) 1/2 a) }, a.e. on. Using triangle inequality and Cauchy-Schwarz inequality, we have d(( u 2 + a 2 ) 1/2 a) = ( u 2 + a 2 ) 1/2( k ) u j du j ( u 2 + a 2 ) 1/2 u du du, where u and du are as in (1.2) and (1.3) respectively. As in (2.7), we obtain d(( v 2 + a 2 ) 1/2 a) dv. (2.7) Therefore, we get dψ max{ du, dv }, a.e. on, (2.8) where dψ is the norm of the cotangent vector dψ, and du and dv are as in (1.3). By definition of φ we have φ [L ()] k and, by definition of ψ we have ψ ( v 2 + a 2 ) 1/2 a.
7 EJDE-2005/?? A PROPERTY OF SOBOLEV SPACES 7 Thus, ψ v, (2.9) where v is as in (1.2). Since v [L ()] k, we have ψ L (). We have already shown that φ [W 1,2 ()] k and ψ W 1,2 (). By Lemma 2.2 (applied to the components ψφ j, 1 j k, of ψφ) we have w = ψφ [W 1,2 ()] k and By (2.10), (2.6) and (2.8), we have a.e. on : dw = (dψ)φ + ψ(dφ) d(ψφ) = (dψ)φ + ψ(dφ). (2.10) dψ φ + ψ dφ (max{ du, dv }) φ + 2( u 2 + a 2 ) 1/2 du ψ max{ du, dv } + 2 du 3 max{ du, dv }, where the third inequality holds since φ 1 and ψ ( u 2 + a 2 ) 1/2 1. This concludes the proof of the Lemma. Lemma 2.8. Let T = (T 1, T 2,..., T k ) and u = (u 1, u 2,..., u k ) be as in the hypotheses of Theorem 1.3. Additionally, assume that u has compact support and u [L ()] k. Then the conclusion of Theorem 1.3 holds. Proof. Since the vector-valued function u = (u 1, u 2,..., u k ) [W 1,2 ()] k is compactly supported, it follows that the functions u j are compactly supported. Thus, using a partition of unity we can assume that u j is supported in a coordinate neighborhood V j. Thus we can use the Friedrichs mollifiers. Let ρ j > 0 and (u j ) ρj := J ρj u, where J ρj denotes the Friedrichs mollifying operator as in Section 5.12 of [2]. Then (u j ) ρ j Cc (), and, as ρ j 0+, we have (u j ) ρ j u j in W 1,2 (); see, for example, Lemma 5.13 in [2]. Thus T j, (u j ) ρj T j, u j, as ρ j 0+, (2.11) where, is as on the right hand side of (1.5). Since (u j ) ρj Cc () and T j L 1 loc (), we have T j, (u j ) ρj = (T j (u j ) ρj ) dν. (2.12) Next, we will show that lim ρ j 0+ (T j (u j ) ρ j ) dν = (T j u j ) dν. (2.13) Since u j L () is compactly supported, by properties of Friedrichs mollifiers (see, for example, the proof of Theorem in [5]) it follows that (i) there exists a compact set K j containing the supports of u j and u ρ j j 0 < ρ j < 1, and (ii) the following inequality holds for all ρ j > 0: for all u ρj j L u j L. (2.14)
8 8 O. ILATOVIC EJDE-2005/?? Since (u j ) ρ j u j in L 2 () as ρ j 0+, after passing to a subsequence we have By (2.14) we have (u j ) ρj u j a.e. on, as ρ j 0 +. (2.15) T j (x)(u j ) ρj (x) T j (x) u j L, a.e. on. (2.16) Since T j L 1 loc (), it follows that T j L 1 (K j ). By (2.15), (2.16) and since T j L 1 (K j ), using dominated convergence theorem, we have lim ρ j 0+ (T j (u j ) ρ j ) dν = lim (T j (u j ) ρ j ) dν = (T j u j ) dν = (T j u j ) dν, ρ j 0+ K j K j and (2.13) is proven. Now, using (2.11), (2.12), (2.13) and the notations (1.4) and (1.5), we get T, u = T j, u j = = = lim T j, (u j ) ρ j ρ j 0+ lim ρ j 0+ This concludes the proof of the Lemma. (T j u j ) dν = (T j (u j ) ρj ) dν (T u) dν. (2.17) Proof of Theorem 1.3. Let u [W 1,2 ()] k. By definition of [W 1,2 ()] k in Section 1, there exists a sequence v m [Cc ()] k such that v m u in [W 1,2 ()] k, as m +. In particular, v m u in [L 2 ()] k, and, hence, we can extract a subsequence, again denoted by v m, such that v m u a.e. on. Define a sequence λ m by λ m := ( u ) 1/2 {( min m 2 u ) 1/2 1 ( m 2 m, v m ) 1/2 1 } m 2, m where v m is the chosen subsequence of v m such that v m u a.e. on, as m +. Clearly, 0 λ m 1. Define w m := λ m u. (2.18) We know that u [W 1,2 ()] k and v m [Cc ()] k. Thus, by Lemma 2.7, for all m = 1, 2, 3,..., we have w m [W 1,2 () L ()] k, and d(w m ) 3 max{ du, d(v m ) }, (2.19) where du is as in (1.2). Furthermore, for all m = 1, 2, 3,..., we have w m (x) u(x), (2.20) where is as in (1.2). Since u [L 2 ()] k, by (2.20) it follows that {w m } is a bounded sequence in [L 2 ()] k. Since v m u in [W 1,2 ()] k, it follows that the sequence {v m } is bounded in [W 1,2 ()] k. In particular, the sequence {d(v m )} is bounded in
9 EJDE-2005/?? A PROPERTY OF SOBOLEV SPACES 9 [L 2 (Λ 1 T )] k. Hence, by (2.19) it follows that {d(w m )} is a bounded sequence in [L 2 (Λ 1 T )] k. Therefore, {w m } is a bounded sequence in [W 1,2 ()] k. By Lemma V.1.4 in [7] it follows that there exists a subsequence of {w m }, which we again denote by {w m }, such that w m converges weakly to some z [W 1,2 ()] k. This means that for every continuous linear functional A [W 1,2 ()] k, we have Since A(w m ) A(z), as m +. [W 1,2 ()] k [L 2 ()] k [W 1,2 ()] k, it follows that w m z in weakly [L 2 ()] k. We will now show that, as m +, w m u in [L 2 ()] k. By definition of w m in (2.18) it follows that w m u a.e. on. Since u [L 2 ()] k, using (2.20) and dominated convergence theorem we get w m u in [L 2 ()] k, as m +. In particular, w m u weakly in [L 2 ()] k. Therefore, by the uniqueness of the weak limit (see, for example, the beginning of Section III.1.6 in [7]), we have z = u. Therefore, w m u weakly in [W 1,2 ()] k. Thus, since T [W 1,2 ()] k, we have T, w m T, u, as m +. (2.21) By the definition of λ m and (2.18) it follows that w m (x) v m (x). (2.22) Since v m [Cc ()] k, by (2.22) it follows that the functions w m have compact support. We have shown earlier that w m [W 1,2 () L ()] k. Thus, by Lemma 2.8, the following equality holds: T, w m = (T w m ) dν. (2.23) Let f be as in the hypotheses of the Theorem. Then T w m = T (λ m u) = λ m (T u) λ m f f. (2.24) By (2.24) it follows that T w m + f 0. Consider the sequence T w m + f. Since f L 1 () and (T w m ) L 1 (), by Fatou s lemma we get lim inf (T m + wm + f ) dν lim inf (T w m + f ) dν. (2.25) m + Since w m u a.e. on as m +, we have T w m T u a.e. on as m +. Thus, by (2.25) we have (T u + f ) dν f dν + lim inf (T w m ) dν, m + and, hence, by (2.23) and (2.21) we have (T u + f ) dν f dν + lim inf (T w m ) dν m + = f dν + lim inf T, m + wm = f dν + T, u.
10 10 O. ILATOVIC EJDE-2005/?? Since f L 1 (), we have (T u + f ) L 1 (), and, hence, (T u) L 1 (). We have T w m = λ m (T u) T u, and by definition of w m, we get, as m +, T w m T u, a.e. on. Using dominated convergence theorem, we get (T w m ) dν = lim m + By (2.26), (2.23) and (2.21), we get T, u = This concludes the proof of the Theorem. (T u) dν. (T u) dν (2.26) Proof of Corollary 1.4. Let T 1 = Re T and T 2 = Im T. Let u 1 = Re u and u 2 = Im u. Then Re T, u = T 1, u 1 + T 2, u 2 and Re(T ū) = T 1 u 1 +T 2 u 2. Thus, Corollary 1.4 follows from Theorem 1.3. References [1] T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer-Verlag, Berlin, [2]. Braverman, O. ilatovic,. Shubin, Essential self-adjointness of Schrödinger type operators on manifolds, Russian ath. Surveys, 57(4) (2002), [3] H. Brézis, F. E. Browder, Sur une propriété des espaces de Sobolev, C. R. Acad. Sci. Paris Sr. A-B, 287, no. 3, (1978), A113 A115. (French). [4] H. Brézis, T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. ath. Pures Appl., 58(9) (1979), [5] G. Friedlander,. Joshi, Introduction to the Theory of Distributions, Cambridge University Press, [6] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, [7] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, Ognjen ilatovic Department of athematics and Statistics, University of North Florida, Jacksonville, FL 32224, USA address: omilatov@unf.edu
SELF-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 informationTHE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS
THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS OGNJEN MILATOVIC Abstract. We consider H V = M +V, where (M, g) is a Riemannian manifold (not necessarily
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 informationOn m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry
On m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry Ognjen Milatovic Department of Mathematics and Statistics University of North Florida Jacksonville, FL 32224 USA. Abstract
More informationMULTIPLE POSITIVE SOLUTIONS FOR KIRCHHOFF PROBLEMS WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 015 015), No. 0, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE POSITIVE SOLUTIONS
More informationEXISTENCE OF WEAK SOLUTIONS FOR A NONUNIFORMLY ELLIPTIC NONLINEAR SYSTEM IN R N. 1. Introduction We study the nonuniformly elliptic, nonlinear system
Electronic Journal of Differential Equations, Vol. 20082008), No. 119, 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) EXISTENCE
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 informationPOTENTIAL LANDESMAN-LAZER TYPE CONDITIONS AND. 1. Introduction We investigate the existence of solutions for the nonlinear boundary-value problem
Electronic Journal of Differential Equations, Vol. 25(25), No. 94, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) POTENTIAL
More informationEXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS
Electronic Journal of Differential Equations, Vol. 017 (017), No. 15, pp. 1 7. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC
More informationNOTE ON THE NODAL LINE OF THE P-LAPLACIAN. 1. Introduction In this paper we consider the nonlinear elliptic boundary-value problem
2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 155 162. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
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 informationMIXED BOUNDARY-VALUE PROBLEMS FOR QUANTUM HYDRODYNAMIC MODELS WITH SEMICONDUCTORS IN THERMAL EQUILIBRIUM
Electronic Journal of Differential Equations, Vol. 2005(2005), No. 123, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) MIXED
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 informationEXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 98, 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) EXISTENCE
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 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 informationWave equation on manifolds and finite speed of propagation
Wave equation on manifolds and finite speed of propagation Ethan Y. Jaffe Let M be a Riemannian manifold (without boundary), and let be the (negative of) the Laplace-Beltrami operator. In this note, we
More informationA Dirichlet problem in the strip
Electronic Journal of Differential Equations, Vol. 1996(1996), No. 10, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.26.103.110 or 129.120.3.113
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 informationPOINTWISE BOUNDS ON QUASIMODES OF SEMICLASSICAL SCHRÖDINGER OPERATORS IN DIMENSION TWO
POINTWISE BOUNDS ON QUASIMODES OF SEMICLASSICAL SCHRÖDINGER OPERATORS IN DIMENSION TWO HART F. SMITH AND MACIEJ ZWORSKI Abstract. We prove optimal pointwise bounds on quasimodes of semiclassical Schrödinger
More informationStationary mean-field games Diogo A. Gomes
Stationary mean-field games Diogo A. Gomes We consider is the periodic stationary MFG, { ɛ u + Du 2 2 + V (x) = g(m) + H ɛ m div(mdu) = 0, (1) where the unknowns are u : T d R, m : T d R, with m 0 and
More informationGENERATORS WITH INTERIOR DEGENERACY ON SPACES OF L 2 TYPE
Electronic Journal of Differential Equations, Vol. 22 (22), No. 89, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GENERATORS WITH INTERIOR
More informationIntegral Representation Formula, Boundary Integral Operators and Calderón projection
Integral Representation Formula, Boundary Integral Operators and Calderón projection Seminar BEM on Wave Scattering Franziska Weber ETH Zürich October 22, 2010 Outline Integral Representation Formula Newton
More informationNONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS INVOLVING EXPONENTIAL CRITICAL GROWTH IN R Introduction
Electronic Journal of Differential Equations, Vol. 014 (014), No. 59, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONLINEAR SCHRÖDINGER
More informationMINIMAL GRAPHS PART I: EXISTENCE OF LIPSCHITZ WEAK SOLUTIONS TO THE DIRICHLET PROBLEM WITH C 2 BOUNDARY DATA
MINIMAL GRAPHS PART I: EXISTENCE OF LIPSCHITZ WEAK SOLUTIONS TO THE DIRICHLET PROBLEM WITH C 2 BOUNDARY DATA SPENCER HUGHES In these notes we prove that for any given smooth function on the boundary of
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 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 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 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 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 informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER NONLINEAR HYPERBOLIC SYSTEM
Electronic Journal of Differential Equations, Vol. 211 (211), No. 78, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
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 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 informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More informationAnalysis in weighted spaces : preliminary version
Analysis in weighted spaces : preliminary version Frank Pacard To cite this version: Frank Pacard. Analysis in weighted spaces : preliminary version. 3rd cycle. Téhéran (Iran, 2006, pp.75.
More informationSolution Sheet 3. Solution Consider. with the metric. We also define a subset. and thus for any x, y X 0
Solution Sheet Throughout this sheet denotes a domain of R n with sufficiently smooth boundary. 1. Let 1 p
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 informationA SHORT PROOF OF INCREASED PARABOLIC REGULARITY
Electronic Journal of Differential Equations, Vol. 15 15, No. 5, pp. 1 9. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu A SHORT PROOF OF INCREASED
More informationEXISTENCE OF STRONG SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS
EXISTENCE OF STRONG SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS Adriana Buică Department of Applied Mathematics Babeş-Bolyai University of Cluj-Napoca, 1 Kogalniceanu str., RO-3400 Romania abuica@math.ubbcluj.ro
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 informationFact Sheet Functional Analysis
Fact Sheet Functional Analysis Literature: Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen. Teubner, 986. Knabner, P., Angermann, L.: Numerik partieller Differentialgleichungen.
More informationRIEMANN-HILBERT PROBLEM FOR THE MOISIL-TEODORESCU SYSTEM IN MULTIPLE CONNECTED DOMAINS
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 310, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu RIEMANN-HILBERT PROBLEM FOR THE MOISIL-TEODORESCU
More informationEXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY
Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN
More informationLIOUVILLE S THEOREM AND THE RESTRICTED MEAN PROPERTY FOR BIHARMONIC FUNCTIONS
Electronic Journal of Differential Equations, Vol. 2004(2004), No. 66, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) LIOUVILLE
More informationASYMPTOTIC BEHAVIOUR OF NONLINEAR ELLIPTIC SYSTEMS ON VARYING DOMAINS
ASYMPTOTIC BEHAVIOUR OF NONLINEAR ELLIPTIC SYSTEMS ON VARYING DOMAINS Juan CASADO DIAZ ( 1 ) Adriana GARRONI ( 2 ) Abstract We consider a monotone operator of the form Au = div(a(x, Du)), with R N and
More informationSTOKES PROBLEM WITH SEVERAL TYPES OF BOUNDARY CONDITIONS IN AN EXTERIOR DOMAIN
Electronic Journal of Differential Equations, Vol. 2013 2013, No. 196, pp. 1 28. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu STOKES PROBLEM
More informationLERAY LIONS DEGENERATED PROBLEM WITH GENERAL GROWTH CONDITION
2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 73 81. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
More informationA VARIATIONAL INEQUALITY RELATED TO AN ELLIPTIC OPERATOR
Proyecciones Vol. 19, N o 2, pp. 105-112, August 2000 Universidad Católica del Norte Antofagasta - Chile A VARIATIONAL INEQUALITY RELATED TO AN ELLIPTIC OPERATOR A. WANDERLEY Universidade do Estado do
More informationLECTURE 22: INTEGRATION ON MANIFOLDS. 1. Orientations
LECTURE 22: INTEGRATION ON ANIFOLDS 1. Orientations Let be a smooth manifold of dimension n, and let ω Ω n () be a smooth n-form. We want to define the integral ω. First assume = R n. In calculus we learned
More informationA LOWER BOUND FOR THE GRADIENT OF -HARMONIC FUNCTIONS Edi Rosset. 1. Introduction. u xi u xj u xi x j
Electronic Journal of Differential Equations, Vol. 1996(1996) No. 0, pp. 1 7. ISSN 107-6691. URL: http://ejde.math.swt.edu (147.6.103.110) telnet (login: ejde), ftp, and gopher access: ejde.math.swt.edu
More informationA note on W 1,p estimates for quasilinear parabolic equations
200-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 2 3. http://ejde.math.swt.edu or http://ejde.math.unt.edu
More informationEXISTENCE OF SOLUTIONS FOR CROSS CRITICAL EXPONENTIAL N-LAPLACIAN SYSTEMS
Electronic Journal of Differential Equations, Vol. 2014 (2014), o. 28, pp. 1 10. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTECE OF SOLUTIOS
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 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 informationSOLUTION OF AN INITIAL-VALUE PROBLEM FOR PARABOLIC EQUATIONS VIA MONOTONE OPERATOR METHODS
Electronic Journal of Differential Equations, Vol. 214 (214), No. 225, pp. 1 1. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SOLUTION OF AN INITIAL-VALUE
More informationWEAK ASYMPTOTIC SOLUTION FOR A NON-STRICTLY HYPERBOLIC SYSTEM OF CONSERVATION LAWS-II
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 94, pp. 1 14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu WEAK ASYMPTOTIC
More informationOn Schrödinger equations with inverse-square singular potentials
On Schrödinger equations with inverse-square singular potentials Veronica Felli Dipartimento di Statistica University of Milano Bicocca veronica.felli@unimib.it joint work with Elsa M. Marchini and Susanna
More informationSecond Order Elliptic PDE
Second Order Elliptic PDE T. Muthukumar tmk@iitk.ac.in December 16, 2014 Contents 1 A Quick Introduction to PDE 1 2 Classification of Second Order PDE 3 3 Linear Second Order Elliptic Operators 4 4 Periodic
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 informationMULTIPLE SOLUTIONS FOR THE p-laplace EQUATION WITH NONLINEAR BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 2006(2006), No. 37, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) MULTIPLE
More informationTHE STOKES SYSTEM R.E. SHOWALTER
THE STOKES SYSTEM R.E. SHOWALTER Contents 1. Stokes System 1 Stokes System 2 2. The Weak Solution of the Stokes System 3 3. The Strong Solution 4 4. The Normal Trace 6 5. The Mixed Problem 7 6. The Navier-Stokes
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 informationGENERIC SOLVABILITY FOR THE 3-D NAVIER-STOKES EQUATIONS WITH NONREGULAR FORCE
Electronic Journal of Differential Equations, Vol. 2(2), No. 78, pp. 1 8. ISSN: 172-6691. UR: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) GENERIC SOVABIITY
More informationAN EXTENSION OF THE LAX-MILGRAM THEOREM AND ITS APPLICATION TO FRACTIONAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 215 (215), No. 95, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu AN EXTENSION OF THE
More informationSOLUTION OF POISSON S EQUATION. Contents
SOLUTION OF POISSON S EQUATION CRISTIAN E. GUTIÉRREZ OCTOBER 5, 2013 Contents 1. Differentiation under the integral sign 1 2. The Newtonian potential is C 1 2 3. The Newtonian potential from the 3rd Green
More informationSYMMETRY AND REGULARITY OF AN OPTIMIZATION PROBLEM RELATED TO A NONLINEAR BVP
lectronic ournal of Differential quations, Vol. 2013 2013, No. 108, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SYMMTRY AND RGULARITY
More informationNONLINEAR ELLIPTIC EQUATIONS WITH MEASURES REVISITED
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURES REVISITED Haïm Brezis (1),(2), Moshe Marcus (3) and Augusto C. Ponce (4) Abstract. We study the existence of solutions of the nonlinear problem (P) ( u + g(u)
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 informationSUPER-QUADRATIC CONDITIONS FOR PERIODIC ELLIPTIC SYSTEM ON R N
Electronic Journal of Differential Equations, Vol. 015 015), No. 17, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SUPER-QUADRATIC CONDITIONS
More informationASYMPTOTIC THEORY FOR WEAKLY NON-LINEAR WAVE EQUATIONS IN SEMI-INFINITE DOMAINS
Electronic Journal of Differential Equations, Vol. 004(004), No. 07, pp. 8. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ASYMPTOTIC
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 informationTheory of PDE Homework 2
Theory of PDE Homework 2 Adrienne Sands April 18, 2017 In the following exercises we assume the coefficients of the various PDE are smooth and satisfy the uniform ellipticity condition. R n is always an
More informationFENGBO HANG AND PAUL C. YANG
Q CURVATURE ON A CLASS OF 3 ANIFOLDS FENGBO HANG AND PAUL C. YANG Abstract. otivated by the strong maximum principle for Paneitz operator in dimension 5 or higher found in [G] and the calculation of the
More informationOn periodic solutions of superquadratic Hamiltonian systems
Electronic Journal of Differential Equations, Vol. 22(22), No. 8, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) On periodic solutions
More informationOn semilinear elliptic equations with measure data
On semilinear elliptic equations with measure data Andrzej Rozkosz (joint work with T. Klimsiak) Nicolaus Copernicus University (Toruń, Poland) Controlled Deterministic and Stochastic Systems Iasi, July
More informationResearch Article Brézis-Wainger Inequality on Riemannian Manifolds
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 7596, 6 pages doi:0.55/2008/7596 Research Article Brézis-Wainger Inequality on Riemannian anifolds Przemysław
More informationExtension and Representation of Divergence-free Vector Fields on Bounded Domains. Tosio Kato, Marius Mitrea, Gustavo Ponce, and Michael Taylor
Extension and Representation of Divergence-free Vector Fields on Bounded Domains Tosio Kato, Marius Mitrea, Gustavo Ponce, and Michael Taylor 1. Introduction Let Ω R n be a bounded, connected domain, with
More informationarxiv: v1 [math.fa] 26 Jan 2017
WEAK APPROXIMATION BY BOUNDED SOBOLEV MAPS WITH VALUES INTO COMPLETE MANIFOLDS PIERRE BOUSQUET, AUGUSTO C. PONCE, AND JEAN VAN SCHAFTINGEN arxiv:1701.07627v1 [math.fa] 26 Jan 2017 Abstract. We have recently
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied athematics http://jipam.vu.edu.au/ Volume 4, Issue 5, Article 98, 2003 ASYPTOTIC BEHAVIOUR OF SOE EQUATIONS IN ORLICZ SPACES D. ESKINE AND A. ELAHI DÉPARTEENT
More informationNONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS
Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University
More informationGLOBAL COMPARISON PRINCIPLES FOR THE p-laplace OPERATOR ON RIEMANNIAN MANIFOLDS
GLOBAL COPARISON PRINCIPLES FOR THE p-laplace OPERATOR ON RIEANNIAN ANIFOLDS ILKKA HOLOPAINEN, STEFANO PIGOLA, AND GIONA VERONELLI Abstract. We prove global comparison results for the p-laplacian on a
More informationPERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 207 222 PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS Fumi-Yuki Maeda and Takayori Ono Hiroshima Institute of Technology, Miyake,
More informationOn the definitions of Sobolev and BV spaces into singular spaces and the trace problem
On the definitions of Sobolev and BV spaces into singular spaces and the trace problem David CHIRON Laboratoire J.A. DIEUDONNE, Université de Nice - Sophia Antipolis, Parc Valrose, 68 Nice Cedex 2, France
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 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 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 informationEXISTENCE AND REGULARITY OF SOLUTIONS FOR STOKES SYSTEMS WITH NON-SMOOTH BOUNDARY DATA IN A POLYHEDRON
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 147, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE AND REGULARITY OF SOLUTIONS FOR
More informationNonlocal Cauchy problems for first-order multivalued differential equations
Electronic Journal of Differential Equations, Vol. 22(22), No. 47, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Nonlocal Cauchy
More informationMATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY
MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY 0.1. Vector Bundles and Connection 1-forms. Let E X be a complex vector bundle of rank r over a smooth manifold. Recall the following abstract
More informationEXISTENCE OF SOLUTIONS TO SYSTEMS OF EQUATIONS MODELLING COMPRESSIBLE FLUID FLOW
Electronic Journal o Dierential Equations, Vol. 15 (15, No. 16, pp. 1 8. ISSN: 17-6691. URL: http://ejde.math.txstate.e or http://ejde.math.unt.e tp ejde.math.txstate.e EXISTENCE OF SOLUTIONS TO SYSTEMS
More informationA non-local problem with integral conditions for hyperbolic equations
Electronic Journal of Differential Equations, Vol. 1999(1999), No. 45, pp. 1 6. ISSN: 17-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login:
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 informationPERIODIC SOLUTIONS FOR NON-AUTONOMOUS SECOND-ORDER DIFFERENTIAL SYSTEMS WITH (q, p)-laplacian
Electronic Journal of Differential Euations, Vol. 24 (24), No. 64, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu PERIODIC SOLUIONS FOR NON-AUONOMOUS
More informationON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM
ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM OLEG ZUBELEVICH DEPARTMENT OF MATHEMATICS THE BUDGET AND TREASURY ACADEMY OF THE MINISTRY OF FINANCE OF THE RUSSIAN FEDERATION 7, ZLATOUSTINSKY MALIY PER.,
More informationGORDON TYPE THEOREM FOR MEASURE PERTURBATION
Electronic Journal of Differential Equations, Vol. 211 211, No. 111, pp. 1 9. ISSN: 172-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GODON TYPE THEOEM FO
More informationNON-EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION SYSTEM WITH NONLOCAL SOURCES
Electronic Journal of Differential Equations, Vol. 2016 (2016, No. 45, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NON-EXTINCTION OF
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 informationA COUNTEREXAMPLE TO AN ENDPOINT BILINEAR STRICHARTZ INEQUALITY TERENCE TAO. t L x (R R2 ) f L 2 x (R2 )
Electronic Journal of Differential Equations, Vol. 2006(2006), No. 5, pp. 6. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A COUNTEREXAMPLE
More informationFUNCTIONAL COMPRESSION-EXPANSION FIXED POINT THEOREM
Electronic Journal of Differential Equations, Vol. 28(28), No. 22, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) FUNCTIONAL
More informationSome Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces
Some Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces S.S. Dragomir Abstract. Some new inequalities for commutators that complement and in some instances improve recent results
More information