Our studies are directed to the existence of weak solutions to a parabolic problem containing a multi-valued term. The problem is formulated in the language of maximal monotone graphs. We assume that the growth and coercivity conditions of a nonlinear term are prescribed by means of time and space dependent $N$--function. This results in formulation of the problem in generalized Musielak-Orlicz spaces. We are using density arguments, hence an important step of the proof is a uniform boundedness of appropriate convolution operators in Musielak-Orlicz spaces. For this purpose we shall need to assume a kind of logarithmic Hölder regularity with respect to $t$ and $x$.
We prove a continuous embedding that allows us to obtain a boundary trace imbedding result for anisotropic Musielak-Orlicz spaces, which we then apply to obtain an existence result for Neumann problems with nonlinearities on the boundary associated to some anisotropic nonlinear elliptic equations in Musielak-Orlicz spaces constructed from Musielak-Orlicz functions on which and on their conjugates we do not assume the $Δ_2$-condition. The uniqueness is also studied.
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.
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.
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.
The Musielak--Orlicz setting unifies the variable exponent, Orlicz, weighted Sobolev, and double-phase spaces. They inherit technical difficulties resulting from general growth and inhomogeneity. In this survey we present an overview of developments of the theory of existence of~PDEs in the setting including reflexive and non-reflexive cases, as well as isotropic and anisotropic ones. Particular attention is paid to problems with data below natural duality in absence of Lavrentiev's phenomenon.
We obtain Musielak Orlicz bumps conditions on a pair of weights for the boundedness of Calderón Zygmund operators and their commutators between variable Lebesgue spaces with different weights. The symbols of the commutators belong to a wider class of functions. We also give Bloom type estimates for commutators of Calderón Zygmund and fractional integral operators in the variable Lebesgue context. The techniques involved in both type of results are related with the theory of sparse domination.
Our purpose is to show the existence of weak solutions for unsteady flow of non-Newtonian incompressible nonhomogeneous, heat-conducting fluids with generalised form of the stress tensor without any restriction on its upper growth. Motivated by fluids of nonstandard rheology we focus on the general form of growth conditions for the stress tensor which makes anisotropic Musielak-Orlicz spaces a suitable function space for the considered problem. We do not assume any smallness condition on initial data in order to obtain long-time existence. Within the proof we use monotonicity methods, integrat…
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 consider a parabolic PDE with Dirichlet boundary condition and monotone operator $A$ with non-standard growth controlled by an $N$-function depending on time and spatial variable. We do not assume continuity in time for the $N$-function. Using an additional regularization effect coming from the equation, we establish the existence of weak solutions and in the particular case of isotropic $N$-function, we also prove their uniqueness. This general result applies to equations studied in the literature like $p(t,x)$-Laplacian and double-phase problems.
In this paper, the extended Musielak-Orlicz-Gauss image problem is studied. Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure $\widetilde{C}_{G,Ψ,λ}(Ω,\cdot)$ of convex body $Ω$ in $\mathbb{R}^{n+1}$ containing the origin (but the origin is not necessary in its interior). In particular, we provide solutions to the extended Musielak-Orlicz-Gauss image problem based on the study of suitably designed parabolic flows, and by the use of approximation technique (for general measures). Our parabolic flows involve two Musielak-Orlicz functions and hence contain many well-stud…
In the present paper we initiate the study of the Musielak-Orlicz-Brunn-Minkowski theory for convex bodies. In particular, we develop the Musielak-Orlicz-Gauss image problem aiming to characterize the Musielak-Orlicz-Gauss image measure of convex bodies. For a convex body $K$, its Musielak-Orlicz-Gauss image measure, denoted by $\widetilde{C}_Θ(K, \cdot)$, involves a triple $Θ=(G, Ψ, λ)$ where $G$ and $Ψ$ are two Musielak-Orlicz functions defined on $S^{n-1}\times (0, \infty)$ and $λ$ is a nonzero finite Lebesgue measure on the unit sphere $S^{n-1}$. Such a measure can be produced by a variati…
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…
We prove existence and uniqueness of renormalized solutions to general nonlinear parabolic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consider \[\partial_t u-\mathrm{div} A(x,\nabla u)= f\in L^1(Ω_T),\] on a Lipschitz bounded domain in $\mathbb{R}^n$. 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 $Δ_2$, nor $\nabla_2$). The condition we impose on $M$ is continuity of log-…
A Class of De Giorgi Type and Hölder Continuity for Some Problems in Musielak-Orlicz-Sobolev Spaces
Higher Institute of Computer Science of Mahdia, University of Monastir（马赫迪高等计算机学院，蒙纳斯提尔大学）；Mathematics Department, Faculty of Sciences, University of Monastir（蒙纳斯提尔大学理学院数学系）
We study a general nonlinear parabolic equation on a Lipschitz bounded domain in $\mathbb{R}^N$, \begin{equation*} \left\{\begin{array}{l l} \partial_t u-\mathrm{div} A(t,x,\nabla u)= f(t,x)&\text{in}\ \ Ω_T,\\ u(t,x)=0 &\ \mathrm{ on} \ (0,T)\times\partialΩ,\\ u(0,x)=u_0(x)&\text{in}\ Ω, \end{array}\right. \end{equation*} with $f\in L^\infty(Ω_T)$ and $u_0\in L^\infty(Ω)$. The growth of the monotone vector field $A$ is controlled by a generalized fully anisotropic $N$-function $M:[0,T)\timesΩ\times\mathbb{R}^N\to[0,\infty)$ inhomogeneous in time and space, and under no growth restrictions on …
In this article, the authors establish a new characterization of the Musielak--Orlicz--Sobolev space on $\mathbb{R}^n$, which includes the classical Orlicz--Sobolev space, the weighted Sobolev space and the variable exponent Sobolev space as special cases, in terms of sharp ball averaging functions. Even in a special case, namely, the variable exponent Sobolev space, the obtained result in this article improves the corresponding result obtained by P. Hästö and A. M. Ribeiro [Commun. Contemp. Math. 19 (2017), 1650022, 13 pp] via weakening the assumption $f\in L^1(\mathbb R^n)$ into $f\in L_{\ma…
We prove in this paper the existence of solutions of nonlinear parabolic problems in inhomogeneous Musielak Orlicz Sobolev spaces, we assume neither a $Δ_2$ nor $\nabla_2$ on the Musielak function $φ$. The main contribution of our work is to prove the existence of entropy solutions without the sign condition on the nonlinearity. The second term $f$ belongs to $L^1(Q)$.
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 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.
For any $p\in(0,\,1]$, let $H^{Φ_p}(\mathbb{R}^n)$ be the Musielak-Orlicz Hardy space associated with the Musielak-Orlicz growth function $Φ_p$, defined by setting, for any $x\in\mathbb{R}^n$ and $t\in[0,\,\infty)$, $$ Φ_{p}(x,\,t):= \begin{cases} \frac{t}{\log(e+t)+[t(1+|x|)^n]^{1-p}} & \qquad \text{when } n(1/p-1)\notin \mathbb{N} \cup \{0\}; \\ \frac{t}{\log(e+t)+[t(1+|x|)^n]^{1-p}[\log(e+|x|)]^p} & \qquad \text{when } n(1/p-1)\in \mathbb{N}\cup\{0\},\\ \end{cases} $$ which is the sharp target space of the bilinear decomposition of the product of the Hardy space $H^p(\mathbb{R}^n)$ and its …
We provide existence and uniqueness of renomalized solutions to a general nonlinear parabolic equation with merely integrable data on a Lipschitz bounded domain in $\mathbb{R}^n$. Namely we study \begin{equation*} \left\{\begin{array}{l } \partial_t u-{\rm div} A(t,x,\nabla u)= f(t,x) \in L^1(Ω_T),\\ u(0,x)=u_0(x)\in L^1(Ω). \end{array}\right. \end{equation*} The growth of the monotone vector field $A$ is assumed to be controlled by a generalized nonhomogeneous and anisotropic $N$-function $M:[0,T)\times Ω\times\mathbb{R}^n \to[0,\infty)$. Existence and uniqueness of renormalized solutions are…
Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ whose heat kernels satisfy the Gaussian upper bound estimates. Assume that the growth function $φ:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$ satisfies that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)\in {\mathbb A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights). Let $H_{φ,\,L}(\mathcal{X})$ be the Musielak-Orlicz-Hardy space defined via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors characteri…
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.
The purpose of this paper is to study a class of double phase problems, with a singular term and a superlinear parametric term on the right-hand side. Using the method of Nehari manifold combined with the fibering maps, we prove that for all small values of the parameter λ > 0, there exist at least two non-trivial positive solutions. Our results extend the previous works Papageorgiou, Repovus, and Vetro [24] and Liu, Dai, Papageorgiou, and Winkert [21], from the case of Musielak-Orlicz Sobolev space, when exponents p and q are constant, to the case of Sobolev-Orlicz spaces with variable expone…
In this paper we introduce a new class of quasilinear elliptic equations driven by the so-called double phase operator with variable exponents. We prove certain properties of the corresponding Musielak-Orlicz Sobolev spaces (an equivalent norm, uniform convexity, Radon-Riesz property with respect to the modular) and the properties of the new double phase operator (continuity, strict monotonicity, (S$_+$)-property). In contrast to the known constant exponent case we are able to weaken the assumptions on the data. Finally we show the existence and uniqueness of corresponding elliptic equations w…
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 $(0,\infty)$.
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 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…
We show local Hölder continuity of quasiminimizers of functionals with non-standard (Musielak--Orlicz) growth. Compared with previous results, we cover more general minimizing functionals and need fewer assumptions. We prove Harnack's inequality and a Morrey type estimate for quasiminimizers. Combining this with Ekeland's variational principle, we obtain local Hölder continuity for $ω$-minimizers.
We provide a direct proof of existence and uniqueness of weak solutions to a broad family of strongly nonlinear elliptic equations with lower order terms. The leading part of the operator satisfies general growth conditions settling the problem in the framework of fully anisotropic and inhomogeneous Musielak--Orlicz spaces generated by an $N$-function $M:Ω\times\mathbb{R}^d\to[0,\infty)$. Neither $\nabla_2$ nor $Δ_2$ conditions are imposed on $M$. Our results cover among others problems with anisotropic polynomial, Orlicz, variable exponent, and double phase growth.
We prove the continuity of bounded solutions for a wide class of parabolic equations with $(p,q)$-growth $$ u_{t}-{\rm div}\left(g(x,t,|\nabla u|)\,\frac{\nabla u}{|\nabla u|}\right)=0, $$ under the generalized non-logarithmic Zhikov's condition $$ g(x,t,{\rm v}/r)\leqslant c(K)\,g(y,τ,{\rm v}/r), \quad (x,t), (y,τ)\in Q_{r,r}(x_{0},t_{0}), \quad 0<{\rm v}\leqslant Kλ(r), $$ $$ \quad \lim\limits_{r\rightarrow0}λ(r)=0, \quad \lim\limits_{r\rightarrow0} \frac{λ(r)}{r}=+\infty, \quad \int_{0} λ(r)\,\frac{dr}{r}=+\infty. $$ In particular, our results cover new cases of double-phase parabolic equat…
In this paper, we study two classes of Kirchhoff type problems set on a double phase framework. That is, the functional space where finding solutions coincides with the Musielak-Orlicz-Sobolev space $W^{1,\mathcal H}_0(Ω)$, with modular function $\mathcal H$ related to the so called double phase operator. Via a variational approach, we provide existence and multiplicity results.
We prove higher integrability of the spatial gradient of weak solutions to parabolic systems with $ϕ$-growth, where $φ=φ(t)$ is a general Orlicz function. The parabolic systems need be neither degenerate nor singular. Our result is a generalized version of the one of J. Kinnunen and J. Lewis [Duke Math. J. 102 (2000), no. 2, 253--271] for the parabolic $p$-Laplace systems.
In this paper we study quasilinear elliptic equations driven by the double phase operator along with a reaction that has a singular and a parametric superlinear term and with a nonlinear Neumann boundary condition of critical growth. Based on a new equivalent norm for Musielak-Orlicz Sobolev spaces and the Nehari manifold along with the fibering method we prove the existence of at least two weak solutions provided the parameter is sufficiently small.
In this paper we study a normalised anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space R^n+1. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Amp`ere type equation. Our argument provides a parabolic proof in the smooth category for the existence of solutions to the Dual Orlicz-Minkowski problem introduced by Zhu, Xing and Ye.
We study nonlinear parabolic PDEs with Orlicz-type growth conditions. The main result gives the existence of a unique solution to the obstacle problem related to these equations. To achieve this we show the boundedness of weak solutions and that a uniformly bounded sequence of weak supersolutions converges to a weak supersolution. Moreover, we prove that if the obstacle is continuous, so is the solution.
This paper addresses the input-to-state stability (ISS) and integral input-to-state stability (iISS) for a class of nonlinear higher dimensional parabolic partial differential equations (PDEs) with different types of boundary disturbances (Robin or Neumann or Dirichlet) from different spaces by means of approximations of Lyapunov functions. Specifically, by constructing approximations of (coercive and non-coercive) ISS Lyapunov functions we establish: (i) the ISS and iISS in $L^1$-norm (and weighted $L^1$-norm) for PDEs with boundary disturbances from $L^q_{loc}(\mathbb{R}_+;L^1(\partialΩ))$-s…
We study the elliptic inclusion given in the following divergence form \begin{align*} & -\mathrm{div}\, A(x,\nabla u) \ni f\quad \mathrm{in}\quad Ω, & u=0\quad \mathrm{on}\quad \partial Ω. \end{align*} As we assume that $f\in L^1(Ω)$, the solutions to the above problem are understood in the renormalized sense. We also assume nonstandard, possibly nonpolynomial, heterogeneous and anisotropic growth and coercivity conditions on the maximally monotone multifunction $A$ which necessitates the use of the nonseparable and nonreflexive Musielak--Orlicz spaces. We prove the existence and uniqueness of…
