相关论文: Calder\'on-Zygmund estimates for higher order syst…

We establish local Calder\'on-Zygmund type estimates for weak solutions to nonlinear parabolic systems with $p$-growth and VMO coefficients. In particular, we prove that if the right-hand side belongs locally to $L^{\mu s}$, where the…

For higher order integral functionals with $p(x)$ growth with respect to the highest order derivative $D^m u$, we prove that $D^m u$ is H\"older continuous on an open subset $\Omega_0 \subset \Omega$ of full Lebesgue- measure, provided that…

In this paper, we prove a higher integrability result for very weak solutions of higher-order elliptic systems involving a double phase operator as the principal part. As a model case, we consider \begin{equation} \int_{\Omega} \left( |D^m…

We consider solutions $u\in W^{1,p}\big(\Omega;\mathbb{R}^{N}\big)$ of the $p$-Laplacian PDE \begin{equation} \nabla\cdot\big(a(x)|Du|^{p-2}Du\big)=0,\notag \end{equation} for $x\in\Omega\subseteq\mathbb{R}^{n}$, where $\Omega$ is open and…

We consider local weak solutions to the fractional $p$-Poisson equation of order $s$, i.e. $\left( - \Delta_p\right)^s u = f$. In the range $p>1$ and $s\in \big(\frac{p-1}{p},1\big)$ we prove Calder\'on & Zygmund type estimates at the…

The paper is concerned with higher order Calderon-Zygmund estimates for the $p$-Laplace equation $$ -\textrm{div}(A(\nabla u)) := -\textrm{div}{(|\nabla u|^{p-2}\nabla u)}=-\textrm{div} F, \qquad 1<p<\infty. $$ We are able to transfer local…

We consider energy solutions of the inhomogeneous parabolic $p$-Laplacien system $\partial_t u-\text{div}(|D u|^{p-2}D u)=-\text{div} g$. We show in the case $p\geq 2$ that if the right hand side $g$ is locally in $L^\infty(\text{BMO})$,…

We consider Calder\'on-Zygmund type estimates for the non-homogeneous $p(\cdot)$-Laplacian system $ -\text{div}(|D u|^{p(\cdot)-2} Du) = -\text{div}(|G|^{p(\cdot)-2} G),$ where $p$ is a variable exponent. We show that $|G|^{p(\cdot)} \in…

We study higher regularity for weak solutions of the $p$-Laplace equation $-\Delta_p u = f$ in a domain $\Omega \subset \mathbb{R}^n$ for $p$ sufficiently close to 2. For $m \ge 3$, assuming that $f$ satisfies suitable Sobolev and H\"older…

We consider local weak solutions to PDEs of the type \[ -\,\mathrm{div}\left((\vert Du\vert-\lambda)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\,\,\,\,\,\,\,\text{in}\,\,\Omega, \] where $1<p<\infty$, $\Omega$ is an open subset of…

This work is concerned with global gradient bounds for a class of divergence-form degenerate elliptic systems with complex-valued coefficients. Notably, the leading coefficients are merely required to be sufficiently small in BMO, which is…

We prove higher integrability of the spatial gradient of weak solutions to parabolic systems with $\phi$-growth, where $\varphi=\varphi(t)$ is a general Orlicz function. The parabolic systems need be neither degenerate nor singular. Our…

We study general parabolic equations of the form $u_t = div A(x,t, u,D u) + div(|F|^{p-2} F)+ f$ whose principal part depends on the solution itself. The vector field $A$ is assumed to have small mean oscillation in $x$, measurable in $t$,…

We consider weak solutions $u:\Omega_{T}\rightarrow\mathbb{R}^{N}$ to parabolic systems of the type \[ u_{t}-\mathrm{div}\,A(x,t,Du)=f \qquad \mathrm{in}\ \Omega_{T}=\Omega\times(0,T), \] where $\Omega$ is a bounded open subset of…

In this paper, we show that $W^{1,p}$ $(1\leq p<\infty)$ weak solutions to divergence form elliptic systems are Lipschitz and piecewise $C^{1}$ provided that the leading coefficients and data are of piecewise Dini mean oscillation, the…

Let $u$ be a weak solution of $ (-\Delta)^m u=f $ with Dirichlet boundary conditions in a smooth bounded domain $\Omega \subset \mathbb{R}^n$. Then, the main goal of this paper is to prove the following a priori estimate: $$…

We study regularity results for nonlinear parabolic systems of $p$-Laplacian type with inhomogeneous boundary and initial data, with $p\in(\frac{2n}{n+2},\infty)$. We show bounds on the gradient of solutions in the Lebesgue-spaces with…

In this paper, we derive suitable optimal $L^p-L^q$ decay estimates, $1\leq p\leq q\leq \infty$, for the solutions to the $\sigma$-evolution equation, $\sigma>1$, with structural damping and power nonlinearity $|u|^{1+\alpha}$ or…

In this paper we prove local gradient estimates and higher differentiability result for the solutions of variational obstacle inequalities \int_\Omega\big<\mathcal{A}(x,u,Du),D(\phi-u)\big>dx\geq \int_\Omega\mathcal{B}(x,u,Du)(\phi-u)dx.…

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded, Lipschitz domain. We consider bounded, weak solutions ($u\in W^{1, 2}\cap L^{\infty}(\Omega;\mathbb{R}^N)$) of the vector-valued, Euler-Lagrange system: \text{div } \big( A(x, u)Du\big)=g(x,…