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We prove global H\"older regularity result for weak solutions $u\in N^{1,p}(\Omega, \mu)$ to a PDE of $p$-Laplacian type with a measure as non-homogeneous term: \[ -\text{div}\!\left( |\nabla u|^{p-2}\nabla u \right)=\overline\nu, \] where…

We investigate quantitative properties of nonnegative solutions $u(x)\ge 0$ to the semilinear diffusion equation $\mathcal{L} u= f(u)$, posed in a bounded domain $\Omega\subset {\mathbb R}^N$ with appropriate homogeneous Dirichlet or outer…

In this paper, we investigate a class of doubly nonlinear evolutions PDEs. We establish sharp regularity for the solutions in H\"older spaces. The proof is based on the geometric tangential method and intrinsic scaling technique. Our…

We establish a gradient estimate for a very weak solution to a quasilinear elliptic equation with a nonstandard growth condition, which is a natural generalization of the $p$-Laplace equation. We investigate the maximum extent for the…

We establish sharp $W^{2,p}$ regularity estimates for viscosity solutions of fully nonlinear elliptic equations under minimal, asymptotic assumptions on the governing operator $F$. By means of geometric tangential methods, we show that if…

We study the existence of weak solutions to the $p$-Navier-Stokes equations with a symmetric $p$-Laplacian on bounded domains. We construct a particular Schauder basis in $W_0^{1,p}(\Omega)$ with divergence free constraint and prove…

This paper is concerned with two properties of positive weak solutions of quasilinear elliptic equations with nonlinear gradient terms. First, we show a Liouville-type theorem for positive weak solutions of the equation involving the…

We study local regularity properties of local minimizer of scalar integral functionals of the form $$\mathcal F[u]:=\int_\Omega F(\nabla u)-f u\,dx$$ where the convex integrand $F$ satisfies controlled $(p,q)$-growth conditions. We…

We consider the numerical solution of the fractional Laplacian of index $s\in(1/2,1)$ in a bounded domain $\Omega$ with homogeneous boundary conditions. Its solution a priori belongs to the fractional order Sobolev space ${\widetilde…

We establish several fine boundary regularity results of weak solutions to non-homogeneous $s$-fractional Laplacian type equations. In particular, we prove sharp Calder\'on-Zygmund type estimates of $u/d^s$ depending on the regularity…

In this paper, we investigate the existence and uniqueness of a non-trivial solution for a class of nonlocal equations involving the fractional $p$-Laplacian operator defined on compact Riemannian manifold, namely,…

We study local regularity properties of local minimizer of scalar integral functionals with controlled $(p,q)$-growth in the two-dimensional plane. We establish Lipschitz continuity for local minimizer under the condition $1<p\leq q<\infty$…

Given $2\leq p<\infty$, $s\in (0, 1)$ and $t\in (1, 2s)$, we establish interior $W^{t,p}$ Calderon-Zygmund estimates for solutions of nonlocal equations of the form \[ \int_{\Omega} \int_{\Omega} K\left (x,|x-y|,\frac{x-y}{|x-y|}\right )…

In this paper, we present a series of Liouville-type theorems for a class of nonhomogeneous quasilinear elliptic equations featuring reactions that depend on the solution and its gradient. Specifically, we investigate equations of the form…

We prove that any nonnegative viscosity solution of the inequality $$(-\Delta_p)^s u(x) \geq u^{t} |\nabla u|^{m}\quad \text{ in }\; \mathbb{R}^N,\; N\geq 2,$$ must be constant. This result holds for parameters $p\in (1, \infty), s\in (0,…

The present paper studies the existence of weak solutions for \begin{equation*} (\mathcal{P}) \left\{\begin{aligned} (-\Delta)^{s_1}_{p_1} u &=\la f_1\,(x,u,v) +g_1(x,u) \,\mbox{ in }\, \Om, \\ (-\Delta)^{s_2}_{p_2} v &=\la f_2\,(x,u,v)…

In this paper we are concerned with the regularity of solutions to a nonlinear elliptic system of $m$ equations in divergence form, satisfying $p$ growth from below and $q$ growth from above, with $p \leq q$; this case is known as $p,…

We consider local weak solutions of widely degenerate elliptic PDEs of the type \begin{equation} \label{equazione mia} \mathrm{div}\Biggl(a(x)(|Du|-1)^{p-1}_+\frac{Du}{|Du|}\Biggr)=b(x,u) \ \ \text{ in }\Omega, \end{equation} where $2\leq…

We show that viscosity solutions to the normalized $p(x)$-Laplace equation coincide with distributional weak solutions to the strong $p(x)$-Laplace equation when $p$ is Lipschitz and $\inf p>1$. This yields $C^{1,\alpha}$ regularity for the…

The aim of this paper is to establish a higher integrability result for very weak solutions of certain parabolic systems whose model is the parabolic $p(x,t)$-Laplacian system. Under assumptions on the exponent function…