Mappings of finite distortion: Compactness.

Iwaniec, Tadeusz; Koskela, Pekka; Onninen, Jani

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Hardy inequalities with non-standard remainder terms

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On Lebesgue points of functions in the Sobolev class $W^{1, n}$ .

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Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

Robert Černý (2014)

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Let n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem $u \in W_{0}^{1} L^{Φ} (Ω) a n d - d i v (Φ^{'} (|\nabla u|) \frac{\nabla u}{|\nabla u|}) + V (x) Φ^{'} (|u|) \frac{u}{|u|} = f (x, u) + μ h (x) i n Ω,$ where ϕ is a Young function such that the space W 01 L Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the...

A-quasiconvexity : relaxation and homogenization

Andrea Braides, Irene Fonseca, Giovanni Leoni (2000)

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Integral representation and $Γ$ -convergence of variational integrals with $p (x)$ -growth

Alessandra Coscia, Domenico Mucci (2002)

ESAIM: Control, Optimisation and Calculus of Variations

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We study the integral representation properties of limits of sequences of integral functionals like $\int f (x, D u) d x$ under nonstandard growth conditions of $(p, q)$ -type: namely, we assume that ${| z |}^{p (x)} \leq f (x, z) \leq {L (1 + | z |}^{p (x)}) .$ Under weak assumptions on the continuous function $p (x)$ , we prove $Γ$ -convergence to integral functionals of the same type. We also analyse the case of integrands $f (x, u, D u)$ depending explicitly on $u$ ; finally we weaken the assumption allowing $p (x)$ to be discontinuous on nice sets.