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Local boundedness of solutions to quasilinear Elliptic Systems
, 2012
"... The mathematical analysis to achieve everywhere regularity in the interior of weak solutions to nonlinear elliptic systems usually starts from their local boundedness. Having in mind De Giorgi’s counterexamples, some structure conditions must be imposed to treat systems of partial differential equ ..."
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Cited by 5 (4 self)
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The mathematical analysis to achieve everywhere regularity in the interior of weak solutions to nonlinear elliptic systems usually starts from their local boundedness. Having in mind De Giorgi’s counterexamples, some structure conditions must be imposed to treat systems of partial differential equations. On the contrary, in the scalar case of a general elliptic single equation a well established theory of regularity exists. In this paper we propose a unified approach to local boundedness of weak solutions to a class of quasilinear elliptic systems, with a structure condition inspired by Ladyzhenskaya–Ural’tseva’s work for linear systems, as well as valid for the general scalar case. Our growth assumptions on the nonlinear quantities involved are new and general enough to include anisotropic systems with sharp exponents and the p, qgrowth case.
Local boundedness of solutions to some anisotropic elliptic systems
 Contemp. Math
"... We consider a map u: Rn → Rm, n,m> 1 solution to a nonlinear system of partial differential equations, or minimizer of a functional of the calculus of variations. It is well known that either the global or the local boundedness of u cannot be obtained through truncation methods. This is due to th ..."
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Cited by 3 (3 self)
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We consider a map u: Rn → Rm, n,m> 1 solution to a nonlinear system of partial differential equations, or minimizer of a functional of the calculus of variations. It is well known that either the global or the local boundedness of u cannot be obtained through truncation methods. This is due to the lack of the maximum principle for general systems. Nevertheless in this paper we present a
Local Boundedness for Vector Valued Minimizers of Anisotropic Functionals
 JOURNAL FOR ANALYSIS AND ITS APPLICATIONS VOLUME XX (20XX), 1–22
"... For variational integrals F(u) = ∫Ω f(x,Du) dx defined on vector valued mappings u: Ω ⊂ Rn → RN, we establish some structure conditions on f that enable us to prove local boundedness for minimizers u ∈ W 1,1(Ω;RN) of F. These structure conditions are satisfied in three remarkable examples: f(x,D ..."
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Cited by 2 (2 self)
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For variational integrals F(u) = ∫Ω f(x,Du) dx defined on vector valued mappings u: Ω ⊂ Rn → RN, we establish some structure conditions on f that enable us to prove local boundedness for minimizers u ∈ W 1,1(Ω;RN) of F. These structure conditions are satisfied in three remarkable examples: f(x,Du) = g(x, Du), f(x,Du) = n∑ j=1 gj(x, uxj ) and f(x,Du) = a(x, (ux1,..., uxn−1)) + b(x, uxn ), for suitable convex functions t → g(x, t), t → gj(x, t), t → a(x, t) and t → b(x, t).
Variational integrals of splittingtype: higher integrability under general growth conditions
, 2007
"... ..."
I E R S I T A I A R A
"... A remark on the regularity of vectorvalued mappings depending on two variables which minimize splittingtype variational integrals ..."
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A remark on the regularity of vectorvalued mappings depending on two variables which minimize splittingtype variational integrals
Twodimensional Variational Problems With A Wide Range Of Anisotropy
, 2010
"... We consider local minimizers u: R2 ⊃ Ω → RM ∫ of the variational integral Ω H(∇u)dx with density H growing at least quadratically and allowing a very large scale of anisotropy. We discuss higher integrability properties of ∇u as well as the differentiability of u in the classical sense. Moreover, a ..."
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We consider local minimizers u: R2 ⊃ Ω → RM ∫ of the variational integral Ω H(∇u)dx with density H growing at least quadratically and allowing a very large scale of anisotropy. We discuss higher integrability properties of ∇u as well as the differentiability of u in the classical sense. Moreover, a Liouvilletype theorem is established.
EXISTENCE OF WEAK SOLUTIONS FOR ELLIPTIC SYSTEMS WITH p, qGROWTH
, 2015
"... Abstract. We consider a nonlinear system of m equations in divergence form and a boundary condition: The functions A α i (x, z) are Hölder continuous with respect to x and We prove the existence of a weak solution u in (ũ + W ..."
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Abstract. We consider a nonlinear system of m equations in divergence form and a boundary condition: The functions A α i (x, z) are Hölder continuous with respect to x and We prove the existence of a weak solution u in (ũ + W