Related papers: New embedding results for double phase problems wi…

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…

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…

This paper deals with the fractional Sobolev spaces $W^{s, p}(\Omega)$, with $s\in (0, 1]$ and $p\in[1,+\infty]$. Here, we use the interpolation results in [4] to provide suitable conditions on the exponents $s$ and $p$ so that the spaces…

In this paper, we consider a new class of multi phase operators with variable exponents, which reflects the inhomogeneous characteristics of hardness changes when multiple different materials are combined together. We at first deal with the…

Sobolev-type embeddings on metric measure spaces encode a subtle interaction between the analytic regularity of functions and the geometry of the underlying domain space. In this paper we develop an embedding theory for variable…

We prove well-posedness results for the Dirichlet problem in $\mathbb{R}^{n}_{+}$ for homogeneous, second order, constant complex coefficient elliptic systems with boundary data in generalized H\"older spaces…

In the present paper, we study a singular double phase variable exponent Dirichlet problem in the setting of a new Musielak-Orlicz Sobolev space with the nonlinearity (the external source) having gradient dependence (so-called convection…

It is well-known that the embedding of the Sobolev space of weakly differentiable functions into H\"{o}lder spaces holds if the integrability exponent is higher than the space dimension. In this paper, the embedding of the Sobolev functions…

The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients…

Let $U$ be a connected open subset of $\mathbb{R}^n$, and let $X=(X_1,X_{2},\ldots,X_m)$ be a system of H\"{o}rmander vector fields defined on $U$. This paper addresses sharp embedding results and geometric inequalities in the generalized…

The Dirichlet problem for a class of stochastic partial differential equations is studied in Sobolev spaces. The existence and uniqueness result is proved under certain compatibility conditions that ensure the finiteness of…

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…

In this paper we will focus on understanding the relation between Sobolev embedding theorems for Haj{\l}asz-Besov spaces defined on a doubling metric measure space $(\Omega,d,\mu)$ and the non-collapsing condition of the measure, i.e. \[…

This short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain…

We investigate asymptotically sharp upper and lower bounds for the approximation numbers of the compact Sobolev embeddings $\overset{\circ}{W}{^m}(\Omega)\hookrightarrow L_2(\Omega)$ and $ W^m(\Omega)\hookrightarrow L_2(\Omega)$, defined on…

Let $\Omega\subseteq \mathbb{R}^{d}$ be open and $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $\Omega$ with $L^{\infty}$ coefficients. Consider the divergence-form operator ${\mathscr L}^{A}=-{\rm…

Under appropriate assumptions on the $N(\Omega)$-fucntion, the De Giorgi process is presented in the framework of Musielak-Orlicz-Sobolev space to prove the H\"{o}lder continuity of fully nonlinear elliptic problems. As the applications,…

In this paper, we study the critical Sobolev embeddings $W^{1,p(x)}(\Omega)\subset L^{p^*(x)}(\Omega)$ for variable exponent Sobolev spaces from the point of view of the $\Gamma$-convergence. More precisely we determine the $\Gamma$-limit…

We establish a Lions-type concentration-compactness principle and its variant at infinity for Musielak-Orlicz-Sobolev spaces associated with a double phase operator with variable exponents. Based on these principles, we demonstrate the…

We prove boundedness, H\"older continuity, Harnack inequality results for local quasiminima to elliptic double phase problems of $p$-Laplace type in the general context of metric measure spaces. The proofs follow a variational approach and…