We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consideron a Lipschitz bounded domain in R N . The growth of the monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N -function M . The approach does not require any particular type of growth condition of M or its conjugate M * (neither ∆ 2 , nor ∇ 2 ). The condition we impose is log-Hölder continuity of M , which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.
We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic N -function, which is not necessarily of power type and need not satisfy the ∆2 nor the ∇2condition. Fully anisotropic, non-reflexive Orlicz-Sobolev spaces provide a natural functional framework associated with these problems. Minimal integrability assumptions are detected on the datum on the right-hand side of the equation ensuring existence and uniqueness of weak solutions. When merely integrable, or even measure, data are allowed, existence of suitably further generalized solutionsin the approximable sense -is established. Their maximal regularity in Marcinkiewicz-type spaces is exhibited as well. Uniqueness of approximable solutions is also proved in case of L 1 -data.
We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised in [40], where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven in [17]. In turn, the a priori estimates found are shown to imply the suitable local Calderón-Zygmund theory for the related class of non-homogeneous, possibly degenerate equations involving discontinuous coefficients. These last results extend to the sub-elliptic setting a few classical non-linear Euclidean results [30,14], and to the non-linear case estimates of the same nature that were available in the sub-elliptic setting only for solutions to linear equations.
We study a general nonlinear parabolic equation on a Lipschitz bounded domain in R N ,in Ω, with f ∈ L ∞ (Ω T ) and u 0 ∈ L ∞ (Ω). The growth of the monotone vector field A is controlled by a generalized fully anisotropic N -function M :inhomogeneous in time and space, and under no growth restrictions on the last variable. It results in the need of the integration by parts formula which has to be formulated in an advanced way. Existence and uniqueness of solutions are proven when the Musielak-Orlicz space is reflexive OR in absence of Lavrentiev's phenomenon. To ensure approximation properties of the space we impose natural assumption that the asymptotic behaviour of the modular function is sufficiently balanced. Its instances are log-Hölder continuity of variable exponent or optimal closeness condition for powers in double phase spaces. The noticeable challenge of this paper is cosidering the problem in non-reflexive and inhomogeneous fully anisotropic space that changes along time.
We prove existence and uniqueness of renormalized solutions to general nonlinear parabolic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consideron a Lipschitz bounded domain in R N . The growth of the weakly monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N -function M . The approach does not require any particular type of growth condition of M or its conjugate M * (neither ∆ 2 , nor ∇ 2 ). The condition we impose on M is continuity of log-Hölder-type, which results in good approximation properties of the space. However, the requirement of regularity can be skipped in the case of reflexive spaces. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments. Uniqueness results from the comparison principle.
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