arXiv subjects

Loïc Constantin

Publications and source records attributed to Loïc Constantin.

5 recordsLinked to original sources

arXiv2026

Trace and Hardy-Sobolev type inequalities and applications to a quasilinear elliptic problem in half-space

In the present paper we are dealing with the following quasilinear elliptic problem: \begin{equation*} \begin{cases} -\mathrm{div}(\rho(x_N) |\nabla u|^{p-2}\nabla u) =a|u|^{s-2}u &\mbox{in }&\ \mathbb{R}^N_+, -|\nabla u|^{p-2}\frac{\partial u}{\partial x_N}=b|u|^{q-2}u&\mbox{on }&\ \mathbb{R}^{N-1}, \end{cases}\ \end{equation*} where $a,b\in \mathbb{R}$, $p,q,s\in(1,\infty)$ and $\rho$ is a continuous positive function on $[0,+\infty)$. We first prove new and sharp embedding results that we establish for the associted weighted energy spaces. In application, we establish existence and regularity of weak solutions to the above problem. We also prove for this problem the nonexistence of nontrivial weak solutions by a new Pohozaev-type identity we obtain. The new results about existence and nonexistence highlight the role of the weight $\rho$ on the solvability of the problem contrasting strongly with those when $\rho$ is constant.

math.AP↗

arXiv2026

A matrix-based spectral method for the numerical approximation of the fractional Laplacian and the fractional $p$-Laplacian of functions defined on $\mathbb R^n$

Given a function $u$ defined on $\mathbb R^n$, its fractional $p$-Laplacian is given by $$(-\Delta)_p^su(\vec x)=C_1(n,s,p)\int_{\mathbb R^n}\frac{|u(\vec x)-u(\vec y)|^{p-2}(u(\vec x)-u(\vec y))}{\|\vec x-\vec y\|_2^{n+sp}}d\vec y,\quad\vec x\in\mathbb R^n,$$where the integral is understood in the principal value sense, $p\in(1,\infty)$, $s\in(0,1)$, and $C_1(n,s,p)$ is a normalization constant. A formally equivalent nonlinear Balakrishnan formulation is given by $$(-\Delta)_p^su(\vec x)=C_4(n,s,p)\int_0^\infty\Delta(t-\Delta)^{-1}\left[\Phi_p(u(\vec x)-u(\cdot))\right](\vec x)\frac{dt}{t^{1-sp/2}},$$ where $C_4(n,s,p)$ is another normalization constant, and $\Phi_p(t)=|t|^{p-2}t$. In this paper, we present a matrix-based spectral method to approximate numerically the fractional Laplacian (i.e., the linear case, where $p = 2$) and the fractional $p$-Laplacian for functions defined on $\mathbb R^n$. Our approach builds on the Balakrishnan representation, where we discretize the 2nd-order derivatives in $\Delta$ using spectrally accurate differentiation matrices. A key advantage is that these matrices can be diagonalized in a well-conditioned manner, enabling a stable and robust numerical scheme that naturally extends to arbitrary spatial dimensions $n$. In particular, this diagonalization allows the fractional operator to act directly on the eigenvalue spectrum, effectively reducing the Balakrishnan integral to an analytical evaluation at the spectral level and thereby avoiding costly multidimensional quadrature. The resulting method also avoids domain truncation and variational formulations, making it both computationally efficient and conceptually straightforward. As a practical application, we simulate the evolution of $$\frac{\partial u}{\partial t}+(-\Delta)^s_pu=0,$$in one and two spatial dimensions, being able to capture the self-similar solutions that arise as $t\to\infty$.

math.NA↗

arXiv2025

Logistic elliptic and parabolic problem for the fractional $p$-Laplacian

In this paper we prove existence, uniqueness of weak solutions of the following nonlocal nonlinear logistic equation \begin{equation*} \begin{cases} (-\Delta)_p^s u_\lambda=\lambda u_\lambda^q - b(x)u_\lambda^r \quad \text{in} \;\Omega,\\ u_\lambda=0 \quad \text{in} \; ( \mathbb{R}^d \backslash \Omega), \\ u_\lambda>0 \text{ in} \; \Omega. \end{cases}\ \end{equation*} We also prove behavior of $u_\lambda$ with respect to $\lambda,$ underlining the effect of the nonlocal operator. We then study the associated parabolic problem, proving local and global existence, uniqueness and global behavior such as stabilization, finite time extinction and blow up.

math.AP↗

arXiv2025

Doubly nonlinear parabolic equation involving a mixed local-nonlocal operator and a convection term

In this paper we study a doubly degenerate parabolic equation involving a convection term and the operator $\mathcal{A}_\mu u:=-\Delta_p u +\mu (-\Delta)^s_q u$ which is a linear combination of the $p$-Laplacian and the fractional $q$-Laplacian, and results in a mixed local-nonlocal nonlinear operator. The problem we study is the following, \begin{equation*} \begin{cases} \partial_t \beta(u)+ \mathcal{A}_\mu u= div (\overset{\to}{f}(u))+g(t,x,u) \quad \text{in} \;Q_T:=(0,T)\times \Omega, u=0 \quad \text{in} \; (0,T)\times (\mathbb{R}^d \backslash \Omega), u(0)=u_0 \text{ in } \Omega. \end{cases}\ \end{equation*} We discuss existence, uniqueness and qualitative behavior of, what we call {\it weak-mild} solutions, that is weak solutions of this problem that when interpreted as $v=\beta(u)$ they are a mild solutions. In particular, we investigate stabilization to steady state, extinction and blow up in finite time and show how the occurrence of such behaviors depend on specific conditions on the nonlinearities $\beta$ (typically of porous media type), $\overset{\to}{f}$ and the source term $g$, and on their relation, in terms of certain regularity and growth conditions.

math.AP↗

arXiv2024

Existence and global behaviour of solutions of a parabolic problem involving the fractional $p$-Laplacian in porous medium

In this paper, we prove the existence and the uniqueness of a weak and mild solution of the following nonlinear parabolic problem involving the porous $p$-fractional Laplacian: \begin{equation*} \begin{cases} \partial_t u+(-\Delta)^s_p(|u|^{m-1}u)=h(t,x,|u|^{m-1}u) & \text{in} \; (0,T)\times \Omega,\\ u=0 & \text{in} \; (0,T) \times \mathbb{R}^d\backslash \Omega, \\ u(0,\cdot)=u_0 & \text{in} \; \Omega . \end{cases}\ \end{equation*} We also study further the the homogeneous case $h(u)=|u|^{q-1}u$ with $q>0$. In particular we investigate global time existence, uniqueness, global behaviour of weak solutions and stabilization.

math.AP↗