We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $E(u,\Omega )={\int }_{\Omega }{\left|\nabla u\right|}^{2}dX+{\mathscr{H}}^n(\{u>0\}\cap \{x_{n+1}=0\}),\Omega \subset {ℝ}^{n+1},$ among all functions $u\ge 0$ which are fixed on $\partial \Omega$.

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 consider an initial-boundary problem for a sixth order nonlinear parabolic equation, which arises in oil-water-surfactant mixtures. Using Schauder type estimates and Campanato spaces, we prove the global existence of classical solutions for the problem in two space dimensions.

The main object of this paper is to study the regularity with respect to the parameter h of solutions of the problem $du/dt+A_h\left(t\right)u\left(t\right)=f_h\left(t\right)$, $u\left(0\right)=u_h^0$. The continuity of u with respect to both h and t has been considered in [6].

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_t=\nabla ·\left(u\nabla {(-\Delta )}^{-s}u\right),0<s<1$. The problem is posed in $\{x\in {ℝ}^n,t\in ℝ\}$ with nonnegative initial data $u(x,0)$ that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and $C^{\alpha }$ regularity of such weak solutions. Finally, we extend the existence...

In this paper, the authors introduce a kind of local Hardy spaces in Rn associated with the local Herz spaces. Then the authors investigate the regularity in these local Hardy spaces of some nonlinear quantities on superharmonic functions on R2. The main results of the authors extend the corresponding results of Evans and Müller in a recent paper.

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 $H^2$.