Related papers: The obstacle problem for a class of degenerate ful…

In this paper we give an overview of some recent and older results concerning free boundary problems governed by elliptic operators.

We investigate here the degenerate bi-harmonic equation: $$\Delta_{m}^2 u=f(x,u)\; \;\;\mbox{in} \O,\quad u = \Delta u = 0\quad \mbox{on }\; \p\Omega,$$ with $m\ge 2,$ and also the degenerate tri-harmonic equation: $$ -\Delta_{m}^3…

In this paper non-transversal intersection of the free and fixed boundary is shown to hold in any dimension for obstacle problems generated by fully nonlinear uniformly elliptic operators. Moreover, $C^1$ regularity results of the free…

We prove some regularity results for a priori bounded local minimizers of non-autonomous integral functionals of the form $$\mathcal{F}(v,\Omega)=\int_\Omega F(x,Dv)dx,$$ under the constraint $v \ge \psi$ a.e. in $\Omega$, where $\psi$ is a…

This paper deals with regularity theory for two-phase free boundary variational problems ruled by degenerate elliptic operators.

In this paper we study the following problem. For any $\ep>0$, take $u^{\ep}$ a solution of, $$ \L u^{\ep}:= {div}\Big(\di\frac {g(|\nabla \uep|)}{|\nabla \uep|}\nabla \uep\Big)=\beta_{\ep}(u^{\ep}),\quad u^{\ep}\geq 0. $$ A solution to…

We consider the numerical solution of the equation - \Delta u - f(u) = g, for the unknown u satisfying Dirichlet conditions in a bounded domain. The nonlinearity f has bounded, continuous derivative. The algorithm uses the finite element…

We consider the nonlinear problem \[(P) \;\; I u=f(x,u) \text{ in $\Omega$,} \;\; u=0 \text{ on $\mathbb{R}^{N}\setminus\Omega$ }\] in an open bounded set $\Omega\subset\mathbb{R}^{N}$, where $I$ is a nonlocal operator which may be…

In this work, we address the questions of existence, uniqueness, and boundary behavior of the positive weak-dual solution of equation $\mathbb{L}_\gamma^s u = \mathcal{F}(u)$, posed in a $C^2$ bounded domain $\Omega \subset \mathbb{R}^N$,…

In this paper we initiate the investigation of free boundary minimization problems ruled by general singular operators with $A_2$ weights. We show existence and boundedness of minimizers. The key novelty is a sharp $C^{1+\gamma}$ regularity…

For an elliptic, semilinear differential operator of the form $S(u) = A : D^2 u + b(x, u , Du)$, consider the functional $E_\infty(u) = \mathop{\mathrm{ess \, sup}}_\Omega |S(u)|$. We study minimisers of $E_\infty$ for prescribed boundary…

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…

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as $$ -F(x,u,Du,D^2u) =\lambda c(x)u+\langle M(x)D u, D u \rangle +h(x) $$ in a bounded domain with a Dirichlet boundary condition, here…

In this manuscript we study geometric regularity estimates for problems driven by fully nonlinear elliptic operators under strong absorption conditions. We establish improved geometric regularity along the free boundary, for a sharp value…

We consider a strongly nonlinear elliptic problem with the homogeneous Dirichlet boundary condition. The growth and the coercivity of the elliptic operator is assumed to be indicated by an inhomogeneous anisotropic $\mathcal{N}$-function.…

In this manuscript, we investigate geometric regularity estimates for problems governed by quasi-linear elliptic models in non-divergence form, which may exhibit either degenerate or singular behavior when the gradient vanishes, under…

In this paper, we consider the obstacle problem for the fractional Laplace operator $(-\Delta)^s$ in the Euclidian space $\mathbb{R}^n$ in the case where $1<s<2$. As first observed in \cite{Y}, the problem can be extended to the upper…

In 1963, Littman, Stampacchia, and Weinberger proved a mean value theorem for elliptic operators in divergence form with bounded measurable coefficients. In the Fermi lectures in 1998, Caffarelli stated a much simpler mean value theorem for…

Let L be a second order, uniformly elliptic operator, and consider the equation L u=f under the homogeneous boundary condition u=0. It is well known that f in C(Om) (Om= Omega) does not guarantee second order derivatives D^2 u in C(Om).…

We show existence of solutions to the least gradient problem on the plane for boundary data in $BV(\partial\Omega)$. We also provide an example of a function $f \in L^1(\partial\Omega) \backslash (C(\partial\Omega) \cup…