Measure data elliptic problems with generalized Orlicz growth
Iwona Chlebicka
备注：
the study is related to arXiv:2008.02227 but is conducted in some other setting and other type of solutions is investigated; our inspiration is arXiv:1701.08970 with integrable data which we allow here to be a measure
We study nonlinear measure data elliptic problems involving the operator exposing generalized Orlicz growth. Our framework embraces reflexive Orlicz spaces, as well as natural variants of variable exponent and double-phase spaces. Approximable and renormalized solutions are proven to exist and coincide for arbitrary measure datum and to be unique when the datum is diffuse with respect to a relevant nonstandard capacity. For justifying that the class of measures is natural, a capacitary characterization of diffuse measures is provided.
We prove Harnack's inequality for bounded weak solutions to quasilinear second order elliptic equations with generalized Orlicz growth conditions. Our approach covers new cases of variable exponent and (p,q) growth conditions.
Under appropriate assumptions on the $N(Ω)$-fucntion, the De Giorgi process is presented in the framework of Musielak-Orlicz-Sobolev space to prove the Hölder continuity of fully nonlinear elliptic problems. As the applications, the Hölder continuity of the minimizers for a class of the energy functionals in Musielak-Orlicz-Sobolev spaces is proved; and furthermore, the Hölder continuity of the weak solutions for a class of fully nonlinear elliptic equations is provided.
In this paper we prove the existence of a nontrivial non-negative radial solution for a quasilinear elliptic problem. Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.
We consider the Dirichlet problem for quasilinear elliptic equations with Musielak-Orlicz (p,q)-growth and non-logarithmic conditions on the coefficients. A sufficient Wiener-type condition for the regularity of a boundary point is established.
We study asymptotic behavior of sub-solutions to non-uniformly elliptic equations with nonstandard growth. In particular, Harnack type inequalities are proved. Our approach gives new results for the cases with (p,q) nonlinearity and generalized Orlicz growth.
We characterize, in the terms of intrinsic Hausdorff measures, the size of~removable sets for Hölder continuous solutions to elliptic equations with Musielak-Orlicz growth. In the general case we provide an elegant form of the measure that captures -- as special cases -- the classical results, slightly refines the ones provided for problems stated in the variable exponent and double phase spaces and essentially improves the known one in the Orlicz case.
We establish some existence and regularity results to the Dirichlet problem, for a class of quasilinear elliptic equations involving a partial differential operator, depending on the gradient of the solution. Our results are formulated in the Orlicz Sobolev spaces and under general growth conditions on the convection term. The sub and supersolutions method is a key tool in the proof of the existence results.
We prove continuity up to the boundary of the minimizer of an obstacle problem and higher integrability of its gradient under generalized Orlicz growth. The result recovers similar results obtained in the special cases of polynomial growth, variable exponent growth and produces new results for Orlicz and double phase growth.
In this paper,we consider the solutions of the non-homogeneous elliptic obstacle problems with Orlicz growth involving measure data. We first establish the pointwise estimates of the approximable solutions to these problems via fractional maximal operators. As a result, we obtain pointwise and oscillation estimates for the gradients of solutions by the non-linear Wolff potentials, and these yield results on $C^{1,α}$-regularity of solutions.
Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established under boundary conditions of Dirichlet, or Neumann, or Robin type. A decisive role in the results is played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients.
We consider nonlinear equations having generalized Orlicz growth (also known as Musielak--Orlicz growth). We prove that if differential operators $\mathcal{A}_i$ converge locally uniformly to an operator $\mathcal{A}$, then the sequence of solutions $(u_i)$ has a subsequence converging to solution $u$ of the limit operator in Sobolev and Hölder norms.
We are concerned with the existence and regularity of the solutions to the Dirichlet problem, for a class of quasilinear elliptic equations driven by a general differential operator, depending on $(x,u,\nabla u)$, and with a convective term $f$. The assumptions on the members of the equation are formulated in terms of Young's functions, therefore we work in the Orlicz-Sobolev spaces. After establishing some auxiliary properties, that seem new in our context, we present two existence and two regularity results. We conclude with several examples.
We prove local $C^{0,α}$- and $C^{1,α}$-regularity for the local solution to an obstacle problem with non-standard growth. These results cover as special cases standard, variable exponent, double phase and Orlicz growth.
We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequality in the associated subunit metric space. For subelliptic operators it is known that the classical Sobolev inequality is sufficient and almost necessary for the Dirichlet problem to be solvable with a quantitative bound on the solution [11]. When the degeneracy is of infinite type, a weaker Orlicz-Sobolev inequality seems to be the right substitute [7]. In this paper we investigate this connection further and reduce the gap betwe…
We obtain Calderón-Zygmund type estimates for parabolic equations with Orlicz growth, where nonlinearities involved in the equations may be discontinuous for the space and time variables. In addition, we consider parabolic systems with the Uhlenbeck structure.
We introduce elliptic and parabolic $\mathcal{B}_{1}$ classes that generalize the well-known $\mathfrak{B}_{p}$ classes of DeGiorgi, Ladyzhenskaya and Ural'tseva with $p>1$. New classes are applied to prove pointwise continuity of solutions of elliptic and parabolic equations with nonstandard growth conditions. Our considerations cover new cases of variable exponent and $(p, q)$-phase growth including the ,,singular-degenerate'' parabolic case $p<2<q$.
We study the following boundary value problem (P)\ \ \ \ \ {-\mathrm{div}(a(|\nabla u|)\nabla u)=f(x,u),\ & in $Ω$, u=0, & on $\partialΩ$} with nonhomogeneous principal part. By assuming the nonlinearity $f(x, t)$ being subcritical growth, some abstract results of problem (P) are obtained: (1) Regularity; (2) Orlicz-Sobolev versus Hölder local minimizer; (3) Strong comparison principle. Applying these abstract results and critical point theory, we prove the existence of multiple solutions of problem (P) in an Orlicz-Sobolev space.
In this paper, we prove the existence of solutions to quasilinear elliptic equations on a bounded domain of $\R^N$ under subcritical Musielak-Orlicz-Sobolev growth. Our proofs rely essentially on Mountain Pass Theorem with corresponding variational techniques. Furthermore, we establish the sharpness of our central assumptions. As far as we know, our approach is new, even for the Orlicz case.
We study unbounded weak supersolutions of elliptic partial differential equations with generalized Orlicz (Musielak--Orlicz) growth. We show that they satisfy the weak Harnack inequality with optimal exponent provided that they belong to a suitable Lebesgue or Sobolev space. Furthermore, we establish the sharpness of our central assumptions.
We study a general nonlinear elliptic equation in the Orlicz setting with data not belonging to the dual of the energy space. We provide several Lorentz-type and Morrey-type estimates for the gradients of solutions under various conditions on the data.
In this paper, we establish pointwise estimates for supersolutions of quasilinear elliptic equations with structural conditions involving a generalized Orlicz growth in terms of a Wolff type potential. As a consequence, under the extra assumption, we obtain that the supersolutions satisfy a Harnack inequality and local Hölder continuity.
This paper is concerned with the multiplicity of nontrivial solutions in an Orlicz-Sobolev space for a nonlocal problem with critical growth, involving N-functions and theory of locally Lispchitz continuous functionals.
We establish pointwise estimates expressed in terms of a nonlinear potential of a generalized Wolff type for $A$-superharmonic functions with nonlinear operator $A:Ω\times\mathbb{R}^n\to\mathbb{R}^n$ having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls bounds from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfies conditions expressed in the natural scales. F…
Under various conditions on the data we analyse how appearence of lower order terms affects the gradient estimates on solutions to a general nonlinear elliptic equation of the form \[-{\rm div}\, a(x,Du)+b(x,u)=μ\] with data $μ$ not belonging to the dual of the natural energy space but to Lorentz/Morrey-type spaces. The growth of the leading part of the operator is governed by a function of Orlicz-type, whereas the lower-order term satisfies the sign condition and is minorized with some convex function, whose speed of growth modulates the regularization of the solutions.
We study solutions to measure data elliptic systems with Uhlenbeck-type structure that involve operator of divergence form, depending continuously on the spacial variable, and exposing doubling Orlicz growth with respect to the second variable. Pointwise estimates for the solutions that we provide are expressed in terms of a nonlinear potential of generalized Wolff type. Not only we retrieve the recent sharp results proven for $p$-Laplace systems, but additionally our study covers the natural scope of operators with similar structure and natural class of Orlicz growth.
In this paper, the semilinear elliptic systems with Dirichlet boundary value are considered \begin{align} \left\{\begin{array}{ll} -Δv=f(u) & \mathrm{in}\ Ω, -Δu=g(v) & \mathrm{in}\ Ω, u=0, \ v=0 & \mathrm{on}\ \partialΩ, \end{array} \right. \end{align} We extend the notion of subcritical growth from polynomial growth to N-function growth. Under N-function growth, nontrivial solutions are obtained via Orlicz-Sobolev spaces and variational methods. It's also noteworthy that the nonlinear term $g(v)$ does not have to satisfy the usual Ambrosetti-Rabinowitz condition. So, in a sense, we enrich re…
We prove a new $\mathcal{A}$-caloric approximation lemma compatible with an Orlicz setting. With this result, we establish a partial regularity result for parabolic systems of the type $$ u_{t}- {\rm div} \,a(Du)=0. $$ Here the growth of $a$ is bounded by the derivative of an $N$-function $φ$. The primary assumption for $φ$ is that $tφ''(t)$ and $φ'(t)$ are uniformly comparable on $(0,\infty)$.
This paper is principally devoted to revisit the remarkable works of Keller and Osserman and generalize some previous results related to the those for the class of quasilinear elliptic problem $$ \left\{ \begin{array}{l} {\rm{div}} \left( ϕ(|\nabla u|)\nabla u\right) = a(x)f(u)\quad \mbox{in } Ω,\\ u\geq0\ \ \mbox{in}\ Ω,\ \ u=\infty\ \mbox{on}\ \partialΩ, \end{array} \right. $$ where either $Ω\subset \mathbb{R}^N$ with $N \geq 1$ is a smooth bounded domain or $Ω= \mathbb{R}^N$. The function $ϕ$ includes special cases appearing in mathematical models in nonlinear elasticity, plasticity, genera…
We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. The motivation of this paper comes from the study on an abstract result in the viewpoint of the fractional maximal distributions and this work also extends some regularity results proved in \cite{PN_dist} by using the weighted fractional maximal distributions (WFMDs). We further investigate a pointwise estimates of the gradient of weak solutions via fractional maximal operators and Riesz potential of data. Moreover, in the setting of the…
We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consider \begin{equation*} -{\rm div} A(x,\nabla u)= f\in L^1(Ω), \end{equation*} on a Lipschitz bounded domain in $\mathbb{R}^N$. 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 $Δ_2$, nor $\nabla_2$). The condition we impose is log-Holder continuity of $M$, whic…
We study the homogenization process for families of strongly nonlinear elliptic systems with the homogeneous Dirichlet boundary conditions. The growth and the coercivity of the elliptic operator is assumed to be indicated by a general inhomogeneous anisotropic $N-$function, which may be possibly also dependent on the spatial variable, i.e., the homogenization process will change the characteristic function spaces at each step. Such a problem is well known and there exists many positive results for the function satisfying $Δ_2$ and $\nabla_2$ conditions an being in addition Hölder continuous wi…
Perron's method and Wiener's criterion have entirely solved the Dirichlet problem for the Laplace equation. Since then, this approach has attracted the attention of many mathematicians for applying these ideas in the more general equations. So, in this paper, we extend the Perron method and the Wiener criterion to the $G(\cdot)$-Laplace equation.
We investigate solutions to nonlinear elliptic Dirichlet problems of the type \[ \left\{\begin{array}{cl} - {\rm div} A(x,u,\nabla u)= μ&\qquad \mathrm{ in}\qquad Ω, u=0 &\qquad \mathrm{ on}\qquad \partialΩ, \end{array}\right. \] where $Ω$ is a bounded Lipschitz domain in $\mathbb{R}^n$ and $A(x,z,ξ)$ is a Carathéodory's function. The growth of~the~monotone vector field $A$ with respect to the $(z,ξ)$ variables is expressed through some $N$-functions $B$ and $P$. We do not require any particular type of growth condition of such functions, so we deal with problems in nonreflexive spaces. When t…
We study the local regularity of $p$-caloric functions or more generally of $ϕ$-caloric functions. In particular, we study local solutions of non-linear parabolic systems with homogeneous right hand side, where the leading terms has Uhlenbeck structure of Orlicz type. This paper closes the gap of [22] where Liebermann proved that if the gradient of a solution is bounded, it is Hölder continuous. The crucial step is a novel local estimates for the gradient of the solutions, which generalize and improve the pioneering estimates of DiBenedetto and Friedman [12,10] for the $p$-Laplace heat equatio…
