We prove boundedness and continuity for solutions to the Dirichlet problem for the equation $$-\mathrm{div}\left(a(x,\nabla u)\right)=h(x,u)+\mu ,\text{in}\Omega \subset {ℝ}^N,$$ where the left-hand side is a Leray-Lions operator from $W_0^{1,p}\left(\Omega \right)$ into $W^{-1,p^{\text{'}}}\left(\Omega \right)$ with $1<p<N$, $h(x,s)$ is a Carathéodory function which grows like ${\left|s\right|}^{p-1}$ and $\mu$ is a finite Radon measure. We prove that renormalized solutions, though not globally bounded, are Hölder-continuous far from the support of $\mu$.

We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of $–{\Delta }_{p}u=g\left(u\right)$ in a smooth bounded domain $\Omega \subset {ℝ}^n$. In particular, we obtain new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^{☆}$ when the domain is strictly convex. More precisely, we prove that $u^{☆}\in L^{\infty }\left(\Omega \right)$ if $n\le p+2$ and $u^{☆}\in L^{\frac{np}{n-p-2}}\left(\Omega \right)\cap W_0^{1,p}\left(\Omega \right)$ if $n>p+2$.

Regularity results for transmission problems in domains with (outgoing) cuspidal points are considered. We prove in some special but generic situations that the solution is piecewise in $H^2$.

In this article we establish the existence of higher order weak derivatives of weak solutions of Dirichlet problem for a class of degenerate elliptic equations.

We study a form of optimal transportation surplus functions which arise in hedonic pricing models. We derive a formula for the Ma–Trudinger–Wang curvature of these functions, yielding necessary and sufficient conditions for them to satisfy (A3w). We use this to give explicit new examples of surplus functions satisfying (A3w), of the form b(x,y) = H(x + y) where H is a convex function on ℝn. We also show that the distribution of equilibrium contracts in this hedonic pricing model is absolutely continuous...

In this paper the Dirichlet problem for a linear elliptic equation in an open, bounded subset of ${\mathbb{R}}^2$ is studied. Regularity properties of the solutions are proved, when the data are $L^1$-functions or Radon measures. In particular sharp assumptions which guarantee the continuity of solutions are given.

We study regularity results for solutions $u\in H{W}^{1,p}\left(\Omega \right)$ to the obstacle problem $${\int }_{\Omega }𝒜(x,{\nabla }_{ℍ}u){\nabla }_{ℍ}(v-u)\mathrm{d}x\ge 0\forall v\in {𝒦}_{\psi ,u}\left(\Omega \right)$$ such that $u\ge \psi$ a.e. in $\Omega$, where ${𝒦}_{\psi ,u}\left(\Omega \right)=\{v\in H{W}^{1,p}\left(\Omega \right):v-u\in H{W}_0^{1,p}\left(\Omega \right)v\ge \psi \text{a.e.}\text{in}\Omega \}$, in Heisenberg groups ${ℍ}^n$. In particular, we obtain weak differentiability in the $T$-direction and horizontal estimates of Calderon-Zygmund type, i.e. $$\begin{array}{ccc}\hfill dT\psi \in H{W}_{\mathrm{loc}}^{1,p}\left(\Omega \right)& \Rightarrow Tu\in L_{\mathrm{loc}}^p\left(\Omega \right),{\left|{\nabla }_{ℍ}\psi \right|}^p\in L_{\mathrm{loc}}^q\left(\Omega \right)\hfill & \hfill \Rightarrow |{\nabla }_{ℍ}{u|}^p\in L_{\mathrm{loc}}^q\left(\Omega \right),\end{array}d$$ where $2<p<4$, $q>1$.