Joao Vitor da Silva | Searcharxiv

arXiv2020

Non-existence of dead cores in fully nonlinear elliptic models

We investigate non-existence of nonnegative dead-core solutions for the problem $$|Du|^γF(x, D^2u)+a(x)u^q = 0 \quad \mbox{in} \quad Ω, \quad u=0 \quad \mbox{ on } \quad \partialΩ.$$ Here $Ω\subset \mathbb{R}^N$ is a bounded smooth domain, $F$ is a fully nonlinear elliptic operator, $a: Ω\to \mathbb{R}$ is a sign-changing weight, $γ\geq 0$, and $0<q<γ+1$. We show that this problem has no non-trivial dead core solutions if either $q$ is close enough to $γ+1$ or the negative part of $a$ is sufficiently small. In addition, we obtain the existence and uniqueness of a positive solution under these conditions on $q$ and $a$. Our results extend previous ones established in the semilinear case, and are new even for the simple model $|D u(x)|^γ \mathrm{Tr}(\mathrm{A}(x) D^2 u(x)) + a(x)u^{q}(x) = 0$, where $\mathrm{A} \in C^0(Ω;Sym(N))$ is a uniformly elliptic and non-negative matrix.

math.AP↗

arXiv2018

An optimization problem with volume constraint with applications to optimal mass transport

In this manuscript we study the following optimization problem with volume constraint: \[ \min\left\{\frac{1}{p}\int_Ω |\nabla v|^pdx- \int_{\partial Ω} gv\,dS \colon v \in W^{1, p} \left(Ω\right), \text{ and } |\{v>0\}| \leq α\right\}. \] Here $g$ is acontinuous function and $α$ is a fixed constant such that $0< α< |Ω|$. Under the assumption that $\displaystyle \int_{\partial Ω} g(x)dS >0$ we prove that a minimizer exists and satisfies $$ \left\{ \begin{array}{ccl} -Δ_p u_p = 0 &\text{in } &\{u_p>0\} \cup \{u_p<0\}, \\[5pt] |\nabla u_p|^{p-2}\frac{\partial u_p}{\partial ν} = g & \text{on } &\partial Ω \cap\partial(\{u_p>0\} \cup \{u_p<0\} ) ,\\[5pt] |\{u_p>0\}| = α. & & \end{array} \right. $$ Next, we analyze the limit as $p\to \infty$. We obtain that any sequence of weak solutions converges, up to a subsequence, $\lim\limits_{p_j \to \infty} u_{p_j}(x)=u_{\infty}(x)$, uniformly in $\overlineΩ$, and uniform limits, $u_\infty$, are solutions to the maximization problem with volume constraint $$ \max\left\{ \int_{\partial Ω} gv\,dS \colon v \in W^{1, \infty} \left(Ω\right), \|\nabla v\|_{L^{\infty}(Ω)}\leq 1 \text{ and }|\{v>0\}| \leq α\right\}. $$ Furthermore, we obtain the limit equation that is verified by $u_\infty$ in the viscosity sense. Finally, it turns out that such a limit variational problem is connected to the Monge-Kantorovich mass transfer problem with the involved measures are supported on $\partial Ω$ and along the limiting free boundary, $\partial \{u_{\infty} \neq 0\}$.

math.AP↗