Displaying similar documents to “Regularity for minimizers of non-autonomous non-quadratic functionals in the case 1 < p < 2 : an a priori estimate”

Integrals with respect to a Radon measure added to area type functionals: semi-continuity and relaxation

Michele Carriero, Antonio Leaci, Eduardo Pascali (1985)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Diamo condizioni sulle funzioni f , g e sulla misura μ affinché il funzionale F ⁢ ( u ) = ∫ Ω f ⁢ ( x , u , D ⁢ u ) ⁢ 𝑑 x + ∫ Ω ¯ g ⁢ ( x , u ) ⁢ 𝑑 μ sia L 1 ⁢ ( Ω ) -semicontinuo inferiormente su W 1 , 1 ⁢ ( Ω ) ∩ C 0 ⁢ ( Ω ¯ ) . Affrontiamo successivamente il problema del rilassamento.

A regularity theory for scalar local minimizers of splitting-type variational integrals

Michael Bildhauer, Martin Fuchs, Xiao Zhong (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of ( p , q ) -growth with exponents p ≤ q &lt; ∞ and show for the scalar case that locally bounded local minimizers are of class C 1 , μ . Note that to our knowledge the only C 1 , μ -results without imposing a relation between p and q concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

Limits of minimum problems for general integral functionals with unilateral obstacles

Gianni Dal Maso (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Se il problema di minimo ( 𝒫 ∞ ) è il limite, in senso variazionale, di una successione di problemi di minimo con ostacoli del tipo min u ≥ ψ h ⁢ ∫ A [ f h ⁢ ( x , u , D ⁢ u ) + b ⁢ ( x , u ) ] ⁢ 𝑑 x , allora ( 𝒫 ∞ ) può essere scritto nella forma min u ⁡ { ∫ A [ f ∞ ⁢ ( x , u , D ⁢ u ) + b ⁢ ( x , u ) ] ⁢ 𝑑 x + ∫ A g ∞ ⁢ ( x , u ~ ⁢ ( x ) ) ⁢ 𝑑 λ ∞ ⁢ ( x ) } dove u ~ è un conveniente rappresentante di u e λ ∞ è una misura non negativa.

Integrals with respect to a Radon measure added to area type functionals: semi-continuity and relaxation

Michele Carriero, Antonio Leaci, Eduardo Pascali (1985)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Diamo condizioni sulle funzioni f , g e sulla misura μ affinché il funzionale F ⁢ ( u ) = ∫ Ω f ⁢ ( x , u , D ⁢ u ) ⁢ d ⁢ x + ∫ Ω ¯ g ⁢ ( x , u ) ⁢ d ⁢ μ sia L 1 ⁢ ( Ω ) -semicontinuo inferiormente su W 1 , 1 ⁢ ( Ω ) ∩ C 0 ⁢ ( Ω ¯ ) . Affrontiamo successivamente il problema del rilassamento.

Sobolev regularity via the convergence rate of convolutions and Jensen’s inequality

Mark A. Peletier, Robert Planqué, Matthias Röger (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We derive a new criterion for a real-valued function u to be in the Sobolev space W 1 , 2 ( ℝ n ) . This criterion consists of comparing the value of a functional ∫ f ( u ) with the values of the same functional applied to convolutions of u with a Dirac sequence. The difference of these values converges to zero as the convolutions approach u , and we prove that the rate of convergence to zero is connected to regularity: u ∈ W 1 , 2 if and only if the convergence is sufficiently fast. We finally apply our criterium to...

On the lower semicontinuity of certain integral functionals

Ennio De Giorgi, Giuseppe Buttazzo, Gianni Dal Maso (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si dimostra che il funzionale ∫ Ω f ⁢ ( u , D ⁢ u ) ⁢ 𝑑 x è semicontinuo inferiormente su W l ⁢ o ⁢ c 1 , 1 ⁢ ( Ω ) , rispetto alla topologia indotta da L l ⁢ o ⁢ c 1 ⁢ ( Ω ) , qualora l’integrando f ⁢ ( s , p ) sia una funzione non-negativa, misurabile in s , convessa in p , limitata nell’intorno dei punti del tipo ( s , 0 ) , e tale che la funzione s ↦ f ⁢ ( s , 0 ) sia semicontinua inferiormente su 𝐑 .

A characterization of Sobolev spaces via local derivatives

David Swanson (2010)

Colloquium Mathematicae

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Let 1 ≤ p < ∞, k ≥ 1, and let Ω ⊂ ℝⁿ be an arbitrary open set. We prove a converse of the Calderón-Zygmund theorem that a function f ∈ W k , p ( Ω ) possesses an L p derivative of order k at almost every point x ∈ Ω and obtain a characterization of the space W k , p ( Ω ) . Our method is based on distributional arguments and a pointwise inequality due to Bojarski and Hajłasz.

On the lower semicontinuity of certain integral functionals

Ennio De Giorgi, Giuseppe Buttazzo, Gianni Dal Maso (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si dimostra che il funzionale ∫ Ω f ⁢ ( u , D ⁢ u ) ⁢ d ⁢ x è semicontinuo inferiormente su W l ⁢ o ⁢ c 1 , 1 ⁢ ( Ω ) , rispetto alla topologia indotta da L l ⁢ o ⁢ c 1 ⁢ ( Ω ) , qualora l’integrando f ⁢ ( s , p ) sia una funzione non-negativa, misurabile in s , convessa in p , limitata nell’intorno dei punti del tipo ( s , 0 ) , e tale che la funzione s ↦ f ⁢ ( s , 0 ) sia semicontinua inferiormente su 𝐑 .

Remarks on the Bourgain-Brezis-Mironescu Approach to Sobolev Spaces

B. Bojarski (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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For a function f ∈ L l o c p ( ℝ ⁿ ) the notion of p-mean variation of order 1, ₁ p ( f , ℝ ⁿ ) is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space W 1 , p ( ℝ ⁿ ) in terms of ₁ p ( f , ℝ ⁿ ) is directly related to the characterisation of W 1 , p ( ℝ ⁿ ) by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.

Partial Boundary Regularity of Solutions of Nonlinear Superelliptic Systems

Christoph Hamburger (2007)

Bollettino dell'Unione Matematica Italiana

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We prove global partial regularity of weaksolutions of the Dirichlet problem for the nonlinear superelliptic system div ⁡ A ⁢ ( x , u , D ⁢ u ) + B ⁢ ( x , u , D ⁢ U ) = 0 , under natural polynomial growth of the coefficient functions A and B . We employ the indirect method of the bilinear form and do not use a Caccioppoli or a reverse Hölder inequality.

Symmetry of minimizers with a level surface parallel to the boundary

Giulio Ciraolo, Rolando Magnanini, Shigeru Sakaguchi (2015)

Journal of the European Mathematical Society

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We consider the functional ℐ Ω ( v ) = ∫ Ω [ f ( | D v | ) - v ] d x , where Ω is a bounded domain and f is a convex function. Under general assumptions on f , Crasta [Cr1] has shown that if ℐ Ω admits a minimizer in W 0 1 , 1 ( Ω ) depending only on the distance from the boundary of Ω , then Ω must be a ball. With some restrictions on f , we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss...

Semicontinuity in L ∞ for polyconvex integrals

Emilio Acerbi, Giuseppe Buttazzo, Nicola Fusco (1982)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Viene studiata la semicontinuità rispetto alla topologia di L ∞ ⁢ ( Ω ; 𝐑 m ) per alcuni funzionali del Calcolo delle Variazioni dipendenti da funzioni a valori vettoriali.

Three Dimensional Vortices in the Nonlinear Wave Equation

Marino Badiale, Vieri Benci, Sergio Rolando (2009)

Bollettino dell'Unione Matematica Italiana

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We prove the existence of rotating solitary waves (vortices) for the nonlinear Klein-Gordon equation with nonnegative potential, by finding nonnegative cylindrical solutions to the standing equation - Δ ⁢ u + μ | y | 2 ⁢ u + λ ⁢ u = g ⁢ ( u ) , u ∈ H 1 ⁢ ( ℝ N ) , ∫ ℝ N u 2 | y | 2 ⁢ 𝑑 x < ∞ , where x = ( y , z ) ∈ ℝ k × ℝ N - k , N > k ≥ 2 , μ > 0 and λ ≥ 0 . The nonnegativity of the potential makes the equation suitable for physical models and guarantees the wellposedness of the corresponding Cauchy problem, but it prevents the use of standard arguments in providing the functional associated to ( † ) with bounded Palais-Smale...

Existence of discontinuous absolute minima for certain multiple integrals without growth properties

Lamberto Cesari (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In the present paper the author discusses certain multiple integrals I ⁢ ( u ) of the calculus of variations satisfying convexity conditions, and no growth property, and the corresponding Serrin integrals ℑ ⁢ ( u ) , to which the existence theorems in [3,4,5] do not apply. However, in the present paper, the integrals I ⁢ ( u ) and ℑ ⁢ ( u ) are reduced to simpler form H ⁢ ( v ) and ℋ ⁢ ( v ) to which the existence theorems above apply. Thus, we derive that I ⁢ ( u ) ≤ ℑ ⁢ ( u ) , H ⁢ ( v ) ≤ ℋ ⁢ ( v ) , we obtain the existence of the absolute minimum for the Serrin forms ℑ ⁢ ( u ) ...

Lower semicontinuous envelopes in W 1 , 1 × L p

Ana Margarida Ribeiro, Elvira Zappale (2014)

Banach Center Publications

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The lower semicontinuity of functionals of the type ∫ Ω f ( x , u , v , ∇ u ) d x with respect to the ( W 1 , 1 × L p ) -weak* topology is studied. Moreover, in absence of lower semicontinuity, an integral representation in W 1 , 1 × L p for the lower semicontinuous envelope is also provided.

Perturbed nonlinear degenerate problems in ℝ N

A. El Khalil, S. El Manouni, M. Ouanan (2009)

Applicationes Mathematicae

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Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ d i v ( x , ∇ u ) + a ( x ) | u | p - 2 u = g ( x ) | u | p - 2 u + h ( x ) | u | s - 1 u in ℝ N ⎨ ⎩ u > 0, l i m | x | → ∞ u ( x ) = 0 , where 1 < p < N; a(x) is assumed to satisfy a coercivity condition; h(x) and g(x) are not necessarily bounded but satisfy some integrability restrictions.

A symmetry problem in the calculus of variations

Graziano Crasta (2006)

Journal of the European Mathematical Society

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We consider the integral functional J ( u ) = ∫ Ω [ f ( | D u | ) − u ] d x , u ∈ W 0 1 , 1 ( Ω ) , where Ω ⊂ ℝ n , n ≥ 2 , is a nonempty bounded connected open subset of ℝ n with smooth boundary, and ℝ ∋ s ↦ f ( | s | ) is a convex, differentiable function. We prove that if J admits a minimizer in W 0 1 , 1 ( Ω ) depending only on the distance from the boundary of Ω , then Ω must be a ball.

Functions with prescribed singularities

Giovanni Alberti, S. Baldo, G. Orlandi (2003)

Journal of the European Mathematical Society

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The distributional k -dimensional Jacobian of a map u in the Sobolev space W 1 , k − 1 which takes values in the sphere S k − 1 can be viewed as the boundary of a rectifiable current of codimension k carried by (part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary M of codimension k can be realized as Jacobian of a Sobolev map valued in S k − 1 . In case M is polyhedral, the...

Existence of discontinuous absolute minima for certain multiple integrals without growth properties

Lamberto Cesari (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

In the present paper the author discusses certain multiple integrals I ⁢ ( u ) of the calculus of variations satisfying convexity conditions, and no growth property, and the corresponding Serrin integrals ℑ ⁢ ( u ) , to which the existence theorems in [3,4,5] do not apply. However, in the present paper, the integrals I ⁢ ( u ) and ℑ ⁢ ( u ) are reduced to simpler form H ⁢ ( v ) and ℋ ⁢ ( v ) to which the existence theorems above apply. Thus, we derive that I ⁢ ( u ) ≤ ℑ ⁢ ( u ) , H ⁢ ( v ) ≤ ℋ ⁢ ( v ) , we obtain the existence of the absolute minimum for the Serrin forms ℑ ⁢ ( u ) ...

Cutting the loss of derivatives for solvability under condition ( Ψ )

Nicolas Lerner (2006)

Bulletin de la Société Mathématique de France

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For a principal type pseudodifferential operator, we prove that condition  ( ψ ) implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from ϵ + 3 / 2 for any ϵ &gt; 0 (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition  ( ψ ) doesimply local solvability with a loss of 1...