Related papers: Uniform bounds for strongly competing systems: the…
We consider a one-phase free boundary problem governed by doubly degenerate fully non-linear elliptic PDEs with non-zero right hand side, which should be understood as an analog (non-variational) of certain double phase functionals in the…
We prove optimal boundary $C^{1,\alpha}$ regularity for viscosity solutions of degenerate fully nonlinear uniformly elliptic equations with oblique boundary conditions and Hamiltonian terms of the form \[ \begin{cases} |Du|^{\gamma}F(D^2 u)…
Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established…
We consider positive solutions to $\displaystyle -\Delta_p u=\frac{1}{u^\gamma}+f(u)$ under zero Dirichlet condition in the half space. Exploiting a prio-ri estimates and the moving plane technique, we prove that any solution is monotone…
In this article, we investigate unique continuation principles for solutions $u$ of uniformly elliptic equations of the form $-\mathrm{div}(A \nabla u) = 0$ when $A$ is less regular than Lipschitz. For general matrices $A$, we prove that…
In the present paper, a systematic study is made of quantitative semicontinuity (a.k.a. Lipschitzian) properties of certain multifunctions, which are defined as a solution map associated to a family of parameterized ``split" feasibility…
In a previous paper we considered a class of infinitely degenerate quasilinear equations and derived a priori bounds for high order derivatives of solutions in terms of the Lipschitz norm. We now show that it is possible to obtain bounds…
In this paper we study, in an open bounded set $\Omega\subset\mathbb R^N$ with Lipschitz boundary $\partial\Omega$, the Dirichlet problem for a nonlinear singular elliptic equation involving the $1$--Laplacian and a total variation term,…
We prove a weak comparison principle in narrow unbounded domains for solutions to $-\Delta_p u=f(u)$ in the case $2<p< 3$ and $f(\cdot)$ is a power-type nonlinearity, or in the case $p>2$ and $f(\cdot)$ is super-linear. We exploit it to…
In this paper, we mainly establish the existence of at least three non-trivial solutions for a class of nonhomogeneous quasilinear elliptic systems with Dirichlet boundary value or Neumann boundary value in a bounded domain…
We consider minimizers $u_\varepsilon$ of the Ginzburg-Landau energy with quadratic divergence penalization on a simply-connected two-dimensional domain $\Omega$. On the boundary, strong tangential anchoring is imposed. We prove that…
We study the existence of fully nontrivial solutions to the system $$-\Delta u_i+ \lambda_iu_i = \sum\limits_{j=1}^\ell \beta_{ij}|u_j|^p|u_i|^{p-2}u_i\ \hbox{in}\ \Omega, \qquad i=1,\ldots,\ell,$$ in a bounded or unbounded domain $\Omega$…
This paper identifies necessary and sufficient conditions for the exactness of penalty functions in optimization problems whose constraint sets are not necessarily bounded. The case where the data of problems is locally Lipschitz,…
In 1995, D. Jerison and C. Kenig in \cite{JK-1995} considered the the inhomogeneous Dirichlet problem $\Delta u= f$ on $\Omega$, $u=0$ on $\partial\Omega$ in Lipschitz domains. One of their main results shows that the $W^{1,p}$ estimate…
We start presenting an $L^{\infty}$-gradient bound for solutions to non-homogeneous $p$-Laplacean type systems and equations, via suitable non-linear potentials of the right hand side. Such a bound implies a Lorentz space characterization…
In this paper, we study semilinear fractional equations $$(-\Delta)^s u(x) = f(u(x))$$ in a half-space and prove that all positive solutions are strictly increasing in the $x_n$-direction. Previous results typically require the solution $u$…
We prove the monotonicity of positive solutions to the problem $-\Delta u = f(u)$ in $\mathbb{R}^N_+ := \{(x',x_N)\in\mathbb{R}^N \mid x_N>0 \}$ under zero Dirichlet boundary condition with a possible singular nonlinearity $f$. In some…
In the theory of non-linear parabolic and elliptic partial differential equations, the notion of maximal regularity plays an essential role in establishing existence, regularity and boundedness of solutions. There is a long history of works…
We study the divergence form second-order elliptic equations with mixed Dirichlet-conormal boundary conditions. The unique $W^{1,p}$ solvability is obtained with $p$ being in the optimal range $(4/3,4)$. The leading coefficients are assumed…
In this paper, we study the existence and the summability of solutions to a Robin boundary value problem whose prototype is the following: $$ \begin{cases} -\text{div}(b(|u|)\nabla u)=f &\text{in }\Omega,\\[.2cm] \displaystyle\frac{\partial…