Global Lipschitz regularity for almost minimizers of asymptotically convex variational problems

We prove global, up to the boundary of a domain ${{\it \Omega}\subset\mathbb {R}^n}We prove global, up to the boundary of a domain W ì \mathbb Rⁿ{{\it \Omega}\subset\mathbb {R}^n}, Lipschitz regularity results for almost minimizers of functionals of the form

u ? òW g(x, u(x), ?u(x)) dx.u \mapsto \int \limits_{\Omega} g(x, u(x), \nabla u(x))\,dx.      3. Lipschitz regularity for scalar minimizers of autonomous simple integrals      Antnio Ornelas 《Journal of Mathematical Analysis and Applications》2004,300(2):889-296 We prove Lipschitz regularity for a minimizer of the integral , defined on the class of the AC functions having x(a)=A and x(b)=B. The Lagrangian may have L(s,) nonconvex (except at ξ=0), while may be non-lsc, measurability sufficing for ξ≠0 provided, e.g., L**() is lsc at (s,0) s. The essential hypothesis (to yield Lipschitz minimizers) turns out to be local boundedness of the quotient φ/ρ() (and not of L**() itself, as usual), where φ(s)+ρ(s)h(ξ) approximates the bipolar L**(s,ξ) in an adequate sense. Moreover, an example of infinite Lavrentiev gap with a scalar 1-dim autonomous (but locally unbounded) lsc Lagrangian is presented.      4. Partial regularity of minimizers of polyconvex variational integrals      Christoph Hamburger 《Calculus of Variations and Partial Differential Equations》2003,18(3):221-241 We prove partial regularity of vector-valued minimizers u of the polyconvex variational integral , where stands for the minors of the gradient Du. For the integrand, we assume f to be a continuous function of class C 2, strictly convex and of polynomial growth in the minors, and g to be a bounded Carathéodory function. We do not employ a Caccioppoli inequality.Received: 19 March 2002, Accepted: 24 October 2002, Published online: 16 May 2003Mathematics Subject Classification (2000): 49N60, 35J50      5. Higher differentiability of minimizers of convex variational integrals      Menita Carozza  Jan Kristensen  Antonia Passarelli di Napoli   《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2011,28(3):781       6. Partial regularity for minimizers of convex integrals with L log L-growth      L. Esposito  G. Mingione 《NoDEA : Nonlinear Differential Equations and Applications》2000,7(1):107-125 We prove -partial regularity of minimizers with for a class of convex integral functionals with nearly linear growth whose model is In this way we extend to any dimension n a previous, analogous, result in [FS] valid only in the case Received December 1998      7. Two-dimensional variational problems with a wide range of anisotropy      M. Fuchs 《Journal of Mathematical Sciences》2011,175(3):375-389 We consider local minimizers u:\mathbbR2 é W? \mathbbRM u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^M} of the variational integral òW H( ?u )dx \int\limits_\Omega {H\left( {\nabla u} \right)dx}      8. Lipschitz and piecewise-C regularity for scalar minimizers of affine simple integrals      António Ornelas 《Journal of Mathematical Analysis and Applications》2004,296(1):21-31 Lipschitz, piecewise-C1 and piecewise affine regularity is proved for AC minimizers of the “affine” integral , under general hypotheses on , , and with superlinear growth at infinity.The hypotheses assumed to obtain Lipschitz continuity of minimizers are unusual: ρ(·) and ?(·) are lsc and may be both locally unbounded (e.g., not in Lloc1), provided their quotient ?/ρ(·) is locally bounded. As to h(·), it is assumed lsc and may take +∞ values freely.      9. Interior gradient bounds for local minimizers of variational integrals under nonstandard growth conditions      D. Apushkinskaya  M. Bildhauer  M. Fuchs 《Journal of Mathematical Sciences》2010,164(3):345-363 Inspired by the work of Marcellini and Papi, we consider local minima u: ℝ n ⊃ Ω → ℝ M of variational integrals òW h( | ?u | ) \int\limits_\Omega {h\left( {\left| {\nabla u} \right|} \right)} dx and prove interior gradient bounds under rather general assumptions on h provided that u is locally bounded. Our requirements on the density h do not involve the dimension n. Bibliography: 18 titles.      10. Partial regularity for minimizers of discontinuous quasi-convex integrals with degeneracy      Verena Bögelein 《Journal of Differential Equations》2012,252(2):1052-1100       11. Higher integrability for minimizers of variational integrals with nonstandard growth      Alessandra Pagano 《Annali dell'Universita di Ferrara》1993,39(1):1-17 We consider a (possibly) vector-valued function u: Ω→R N, Ω⊂R n, minimizing the integral , whereD iu=∂u/∂x i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D 1u,…,Dn−1u∈Lq, under suitable assumptions ona i(x). Sunto Consideriamo una funzione u: Ω→R N, Ω⊂R n che minimizzi l'integrale , doveD iu=∂u/∂xi, o un funzionale con un comportamento simile; sotto opportune ipotesi sua i(x), dimostriamo la seguente maggiore integrabilità perDu:D 1u,…,Dn−1uεLq.       12. Boundary regularity for almost minimizers of quasiconvex variational problems      Manfred Kronz 《NoDEA : Nonlinear Differential Equations and Applications》2005,12(3):351-382 We consider boundary regularity for almost minimizers of quasiconvex variational integrals with polynomial growth of order p ≥ 2, and obtain a general criterion for an almost minimizer to be regular in the neighbourhood of a given boundary point. Combined with existing results on interior partial regularity, the proof yields directly the optimal regularity for an almost minimizer in this neighbourhood.      13. Global regularity for almost minimizers of nonconvex variational problems      M. Foss 《Annali di Matematica Pura ed Applicata》2008,187(2):263-321 We prove some global, up to the boundary of a domain $\Omega \subset {\mathbb{R}}^{n}We prove some global, up to the boundary of a domain , continuity and Lipschitz regularity results for almost minimizers of functionals of the form The main assumption for g is that it be asymptotically convex with respect its third argument. For the continuity results, the integrand is allowed to have some discontinuous behavior with respect to its first and second arguments. For the global Lipschitz regularity result, we require g to be H?lder continuous with respect to its first two arguments.       14. Partial regularity of minimizers of quasiconvex integrals with subquadratic growth      Menita Carozza  Nicola Fusco  Giuseppe Mingione 《Annali di Matematica Pura ed Applicata》1998,175(1):141-164 We prove partial regularity for minimizers of quasiconvex integrals of the form dx where the integral F() has subquadratic growth, ie .Research supported by MURST, Gruppo Nazionale 40%.      15. Partial regularity for minima of variational integrals      Mariano Giaquinta  Per-Anders Ivert 《Arkiv f?r Matematik》1987,25(1-2):221-229       16. Partial regularity for minima of variational integrals      Mariano Giaquinta  Per-Anders Ivert 《Arkiv f?r Matematik》1987,25(1):221-229       17. Higher differentiability of minimizers of variational integrals with variable exponents      Flavia Giannetti  Antonia Passarelli di Napoli 《Mathematische Zeitschrift》2015,280(3-4):873-892       18. Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth      Frank?DuzaarEmail author  Joseph F.?Grotowski  Manfred?Kronz 《Annali di Matematica Pura ed Applicata》2005,184(4):421-448 We prove a small excess regularity theorem for almost minimizers of a quasi-convex variational integral of subquadratic growth. The proof is direct, and it yields an optimal modulus of continuity for the derivative of the almost minimizer. The result is new for general almost minimizers, and in the case of absolute minimizers it considerably simplifies the existing proof. Mathematics Subject Classification (2000) 49N60, 26B25      19. Existence of vector minimizers for nonconvex 1-dim integrals with almost convex Lagrangian      Clara Carlota  Antnio Ornelas 《Journal of Differential Equations》2007,243(2):414-426 This paper proves new results of existence of minimizers for the nonconvex integral , among the AC functions with x(a)=A, x(b)=B. Our Lagrangian L() is e.g. lsc with superlinear growth, assuming +∞ values freely. We replace convexity by almost convexity, a hypothesis which in the radial superlinear case L(s,ξ)=f(s,|ξ|) is automatically satisfied provided f(s,) is convex at zero.      20. The partial regularity for minimizers of splitting type variational integrals under general growth conditions II. The nonautonomous case      D. Breit 《Journal of Mathematical Sciences》2010,166(3):259-281

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Lipschitz regularity for scalar minimizers of autonomous simple integrals

We prove Lipschitz regularity for a minimizer of the integral , defined on the class of the AC functions having x(a)=A and x(b)=B. The Lagrangian may have L(s,) nonconvex (except at ξ=0), while may be non-lsc, measurability sufficing for ξ≠0 provided, e.g., L^**() is lsc at (s,0) s. The essential hypothesis (to yield Lipschitz minimizers) turns out to be local boundedness of the quotient φ/ρ() (and not of L^**() itself, as usual), where φ(s)+ρ(s)h(ξ) approximates the bipolar L^**(s,ξ) in an adequate sense. Moreover, an example of infinite Lavrentiev gap with a scalar 1-dim autonomous (but locally unbounded) lsc Lagrangian is presented.     

Partial regularity of minimizers of polyconvex variational integrals

We prove partial regularity of vector-valued minimizers u of the polyconvex variational integral , where stands for the minors of the gradient Du. For the integrand, we assume f to be a continuous function of class C ², strictly convex and of polynomial growth in the minors, and g to be a bounded Carathéodory function. We do not employ a Caccioppoli inequality.Received: 19 March 2002, Accepted: 24 October 2002, Published online: 16 May 2003Mathematics Subject Classification (2000): 49N60, 35J50     

Partial regularity for minimizers of convex integrals with L log L-growth

We prove -partial regularity of minimizers with for a class of convex integral functionals with nearly linear growth whose model is In this way we extend to any dimension n a previous, analogous, result in [FS] valid only in the case Received December 1998     

Two-dimensional variational problems with a wide range of anisotropy

We consider local minimizers u:\mathbbR² é W? \mathbbR^M u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^M} of the variational integral

Lipschitz and piecewise-C regularity for scalar minimizers of affine simple integrals

Lipschitz, piecewise-C¹ and piecewise affine regularity is proved for AC minimizers of the “affine” integral , under general hypotheses on , , and with superlinear growth at infinity.The hypotheses assumed to obtain Lipschitz continuity of minimizers are unusual: ρ(·) and ?(·) are lsc and may be both locally unbounded (e.g., not in L_loc¹), provided their quotient ?/ρ(·) is locally bounded. As to h(·), it is assumed lsc and may take +∞ values freely.     

Interior gradient bounds for local minimizers of variational integrals under nonstandard growth conditions

Inspired by the work of Marcellini and Papi, we consider local minima u: ℝ ⁿ ⊃ Ω → ℝ ^M of variational integrals ò_W h( | ?u | ) \int\limits_\Omega {h\left( {\left| {\nabla u} \right|} \right)} dx and prove interior gradient bounds under rather general assumptions on h provided that u is locally bounded. Our requirements on the density h do not involve the dimension n. Bibliography: 18 titles.     

Higher integrability for minimizers of variational integrals with nonstandard growth

We consider a (possibly) vector-valued function u: Ω→R ^N, Ω⊂R ⁿ, minimizing the integral , whereD _iu=∂u/∂x _i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D ₁u,…,D_n−1u∈L^q, under suitable assumptions ona _i(x).

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Sunto Consideriamo una funzione u: Ω→R ^N, Ω⊂R ⁿ che minimizzi l'integrale , doveD _iu=∂u/∂x_i, o un funzionale con un comportamento simile; sotto opportune ipotesi sua _i(x), dimostriamo la seguente maggiore integrabilità perDu:D ₁u,…,D_n−1uεL^q.

     

Boundary regularity for almost minimizers of quasiconvex variational problems

We consider boundary regularity for almost minimizers of quasiconvex variational integrals with polynomial growth of order p ≥ 2, and obtain a general criterion for an almost minimizer to be regular in the neighbourhood of a given boundary point. Combined with existing results on interior partial regularity, the proof yields directly the optimal regularity for an almost minimizer in this neighbourhood.     

Global regularity for almost minimizers of nonconvex variational problems

We prove some global, up to the boundary of a domain $\Omega \subset {\mathbb{R}}^{n}We prove some global, up to the boundary of a domain , continuity and Lipschitz regularity results for almost minimizers of functionals of the form The main assumption for g is that it be asymptotically convex with respect its third argument. For the continuity results, the integrand is allowed to have some discontinuous behavior with respect to its first and second arguments. For the global Lipschitz regularity result, we require g to be H?lder continuous with respect to its first two arguments.      

Partial regularity of minimizers of quasiconvex integrals with subquadratic growth

We prove partial regularity for minimizers of quasiconvex integrals of the form dx where the integral F() has subquadratic growth, ie .Research supported by MURST, Gruppo Nazionale 40%.     

Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth

We prove a small excess regularity theorem for almost minimizers of a quasi-convex variational integral of subquadratic growth. The proof is direct, and it yields an optimal modulus of continuity for the derivative of the almost minimizer. The result is new for general almost minimizers, and in the case of absolute minimizers it considerably simplifies the existing proof. Mathematics Subject Classification (2000) 49N60, 26B25     

Existence of vector minimizers for nonconvex 1-dim integrals with almost convex Lagrangian

This paper proves new results of existence of minimizers for the nonconvex integral , among the AC functions with x(a)=A, x(b)=B. Our Lagrangian L() is e.g. lsc with superlinear growth, assuming +∞ values freely. We replace convexity by almost convexity, a hypothesis which in the radial superlinear case L(s,ξ)=f(s,|ξ|) is automatically satisfied provided f(s,) is convex at zero.