arXiv subjects
Publications and source records attributed to Anup Biswas.
In this article, we establish Schauder-type estimates for the logarithmic Laplacian. We show that for a $\kappa$-H\"older inhomogeneous term, the solution is also $\kappa$-H\"older in the interior. In fact, the interior regularity slightly exceeds $\kappa$-H\"older smoothness up to a logarithmic correction. Additionally, we prove a Harnack inequality for non-negative solutions.
In this article, we study the equation $$ (-\Delta_p)^s u = f(u) $$ in a bounded domain $\Omega\subset \mathbb{R}^n$, where $n\geq 2$, $p>2$, and $f$ is locally Lipschitz. We establish a strong comparison principle in a fairly general setting and use it to derive symmetry results for positive $C^1$ solutions satisfying Dirichlet boundary conditions. We also show that the $C^1$ regularity assumption is indeed satisfied for $p\in \left[2,\frac{2}{1-s}\right)$.
In this article, we study nonlinear nonlocal equations with coercive gradient nonlinearity of the form \[ (-\Delta_p)^s u(x) + H(x, \nabla u) = f, \] where $f$ is Lipschitz continuous. We show that any viscosity solution $u$ is locally Lipschitz continuous, provided \[ p \in \left(1, \frac{2}{1-s}\right) \cup (1, m+1). \] We also establish H\"older continuity of subsolutions. Furthermore, in the case $f=0$ and $H$ is independent of $x$, we prove that the equation admits only the trivial solution in the class of bounded solutions, for all $m, p \in (1,\infty)$.
We establish the local $C^{1, \alpha}$ regularity of minimizers for functionals of the form $$w\to \int_{\Omega}(|\nabla w|^p-fw) dx + \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{|w(x)-w(y)|^q}{|x-y|^{n+sq}}dx\, dy,$$ where $s \in (0, 1)$, $1 < p \leq sq$, and $f \in L^\infty(\Omega)$. This result complements the work of De Filippis and Minigione in \cite{DFM}, thereby completing the proof of $C^{1,\alpha}$ regularity for all $p, q \in (1, \infty)$ and $s \in (0, 1)$ with locally bounded source term.
We prove that any nonnegative viscosity solution of the inequality $$(-\Delta_p)^s u(x) \geq u^{t} |\nabla u|^{m}\quad \text{ in }\; \mathbb{R}^N,\; N\geq 2,$$ must be constant. This result holds for parameters $p\in (1, \infty), s\in (0, 1)$, $t, m\geq 0$, satisfying $$t (N-sp) + m(N-(sp-p+1)) < N(p-1),$$ with the additional condition that either $m\leq p-1$ if $p-1<sp$, or $m<sp$ if $p-1\geq sp$.
In this paper, we establish several Liouville-type theorems for a class of nonhomogenenous quasilinear inequalities. In the first part, we prove various Liouville results associated with nonnegative solutions to \begin{equation*}\tag{$P_s$} -\Delta_p u-\Delta_q u\geq u^{s-1} \, \text{ in }\, \Omega, \end{equation*} where $1 1$ and $\Omega$ is any exterior domain of $\mathbb{R}^N$. In particular, we prove that for $q q_*$, where $q_*=\frac{q(N-1)}{N-q}$ is the Serrin exponent for the $q$-Laplacian. Further, we show that when $s=q_*$ and $p 1$. In the second part, we consider the inequality \begin{equation*}\tag{$P_{sm}$} -\Delta_p u-\Delta_q u \geq u^s |\nabla u|^m \quad \text{ in }\mathbb{R}^N, \end{equation*} where $1 q$ and $s, \, m\geq 0$. We prove that, for $\{0\leq m\leq q-1\}\cup\{m>p-1\}$, the only positive solution to $(P_{sm})$ is constant, provided $s(N-q)+m(N-1)<N(q-1)$. This, in particular, proves that if $\Omega=\mathbb{R}^N$ then any nonnegative solution to $(P_s)$ with $1<q<N$ and $1<s<q_*$ is the trivial solution. To prove Liouville in the range $0\leq m<q-1$, we first prove an almost optimal lower estimate of any nonnegative supersolution of $(P_{sm})$ and then leveraging this estimate we prove Liouville result. To the best of our knowledge, this technique is completely new and provides an alternative approach to the capacity method of Mitidieri-Pohozaev provided higher regularity is available.
In this paper, we present a series of Liouville-type theorems for a class of nonhomogeneous quasilinear elliptic equations featuring reactions that depend on the solution and its gradient. Specifically, we investigate equations of the form $-\Delta_p u - \Delta_q u = f(u,\nabla u)$ with $p > q > 1$, where the nonlinearity $f$ takes forms such as $u^s|\nabla u|^m$ or $u^s + M|\nabla u|^m$ ($s, m\geq 0$). Our approach is twofold. For cases where the reaction term satisfies $|f(u,\nabla u)|\leq g(u)|\nabla u|^m$ with $m>q$ and $g$ is continuous, we prove that every bounded solution (without sign restriction) in $\mathbb{R}^N$ is constant by means of an Ishii-Lions type technique. In the remaining scenarios, we turn to the Bernstein method. The application of this method to the nonhomogeneous operator requires a nontrivial adaptation, as, roughly speaking, constant coefficients are replaced by functions that may not be bounded from above, which enables us to establish a crucial a priori estimate for the gradient of solutions in any domain $\Omega$. This estimate, in turn, implies the desired Liouville properties on the entire space $\mathbb{R}^N$. As a consequence, we have fully extended Lions Liouville-type result for the Hamilton-Jacobi equation to the $(p,q)$-Laplacian setting, while for the $(p,q)$ generalized Lane-Emden equation, we provide an initial contribution in the direction of the classical result by Gidas and Spruck for $p=q=2$, as well as that of Serrin and Zou for $p=q$. To the best of our knowledge, this is the first paper which studies Liouville properties for equations with nonhomogeneous operator involving source gradient terms.
We prove a quantitative H\"{o}lder continuity result for viscosity solutions to the equation $$ (-\Delta_p)^{s}u(x) + {\rm PV} \int_{\mathbb{R}^n} |u(x)-u(x+z)|^{q-2}(u(x)-u(x+z))\frac{\xi(x,z)}{|z|^{n+ tq}} dz=f \quad \text{in}\; B_2, $$ where $t, s\in (0, 1), 1 2, \\ \min\{1, \frac{sp+\alpha\wedge\beta}{p-1}\} & \text{for}\; p\in (1, 2]. \end{array} \right. \] Moreover, if $\min\{\frac{sp+\alpha\wedge\beta}{p-1}, \frac{sp}{p-2}\}>1$ when $p>2$, or $\frac{sp+\alpha\wedge\beta}{p-1}>1$ when $p\in (1, 2]$, the solution is locally Lipschitz. This extends the result of [20] to the case of H\"{o}lder continuous modulating coefficients. Additionally, due to the equivalence between viscosity and weak solutions, our result provides a local Lipschitz estimate for weak solutions of $(-\Delta_p)^{s}u(x)=0$ provided either $p\in (1, 2]$ or $sp>p-2$ when $p>2$, thereby improving recent works [9, 10, 24].
In this article, we investigate the H\"{o}lder regularity of the fractional $p$-Laplace equation of the form $(-\Delta_p)^s u=f$ where $p>1, s\in (0, 1)$ and $f\in L^\infty_{\rm loc}(\Omega)$. Specifically, we prove that $u\in C^{0, \gamma_\circ}_{\rm loc}(\Omega)$ for $\gamma_\circ=\min\{1, \frac{sp}{p-1}\}$, provided that $\frac{sp}{p-1}\neq 1$. In particular, it shows that $u$ is locally Lipschitz for $\frac{sp}{p-1}>1$. Moreover, we show that for $\frac{sp}{p-1}=1$, the solution is locally Lipschitz, provided that $f$ is locally H\"{o}lder continuous. Additionally, we discuss further regularity results for the fractional double-phase problems.
Given a metric measure space $(\mathcal{X}, d, \mu)$ satisfying the volume doubling condition, we consider a semigroup $\{S_t\}$ and the associated heat operator. We propose general conditions on the heat kernel so that the solutions of the associated heat equations attain the initial data pointwise. We demonstrate that these conditions are satisfied by a broad class of operators, including the Laplace operators perturbed by a gradient, fractional Laplacian, mixed local-nonlocal operators, Laplacian on Riemannian manifolds, Dunkl Laplacian and many more. In addition, we consider the Laplace operator in $\mathbb{R}^n$ with the Hardy potential and establish a characterization for the pointwise convergence to the initial data. We also prove similar results for the nonhomogeneous equations and showcase an application for the power-type nonlinearities.
In this article we prove the Pohozaev identity for the semilinear Dirichlet problem of the form $-\Delta u + a(-\Delta)^s u = f(u)$ in $\Omega$, and $u=0$ in $\Omega^c$, where $a$ is a non-negative constant and $\Omega$ is a bounded $C^2$ domain. We also establish similar identity for systems of equations. As applications of this identity, we deduce a unique continuation property of eigenfunctions and also the nonexistence of nontrivial solutions in star-shaped domains under suitable condition on $f$.
In this article we consider a large family of nonlinear nonlocal equations involving gradient nonlinearity and provide a unified approach, based on the Ishii-Lions type technique, to establish Liouville properties of the solutions. We also answer an open problem raised by [24]. Some applications to regularity issues are also studied.
We study the existence-uniqueness of solution $(u, λ)$ to the ergodic Hamilton-Jacobi equation $$(-Δ)^s u + H(x, \nabla u) = f-λ\quad \text{in}\; \mathbb{R}^d,$$ and $u\geq 0$, where $s\in (\frac{1}{2}, 1)$. We show that the critical $λ=λ^*$, defined as the infimum of all $λ$ attaining a non-negative supersolution, attains a nonnegative solution $u$. Under suitable conditions, it is also shown that $λ^*$ is the supremum of all $λ$ for which a non-positive subsolution is possible. Moreover, uniqueness of the solution $u$, corresponding to $λ^*$, is also established. Furthermore, we provide a probabilistic characterization that determines the uniqueness of the pair $(u, λ^*)$ in the class of all solution pair $(u, λ)$ with $u\geq 0$. Our proof technique involves both analytic and probabilistic methods in combination with a new local Lipschitz estimate obtained in this article.
In this article we consider a class of non-degenerate elliptic operators obtained by superpositioning the Laplacian and a general nonlocal operator. We study the existence-uniqueness results for Dirichlet boundary value problems, maximum principles and generalized eigenvalue problems. As applications to these results, we obtain Faber-Krahn inequality and a one-dimensional symmetry result related to the Gibbons' conjecture. The latter results substantially extend the recent results of Biagi et.\ al. [7,9] who consider the operators of the form $-Δ+ (-Δ)^s$ with $s\in (0, 1)$.
In this article we consider a class of nonlinear integro-differential equations of the form $$\inf_{τ\in\mathcal{T}} \bigg\{\int_{\mathbb{R}^d} (u(x+y)+u(x-y)-2u(x))\frac{k_τ(x,y)}{|y|^{d+2s}} \,dy+ b_τ(x) \cdot \nabla u(x)+g_τ(x) \bigg\}-λ^*=0\quad \text{in} \hspace{2mm} \mathbb{R}^d,$$ where $0<λ(2-2s)\leq k_τ\leq Λ(2-2s)$ , $s\in (\frac{1}{2},1)$. The above equation appears in the study of ergodic control problems in $\mathbb{R}^d$ when the controlled dynamics is governed by pure-jump Lévy processes characterized by the kernels $k_τ\,|y|^{-d-2s}$ and the drift $b_τ$. Under a Foster-Lyapunov condition, we establish the existence of a unique solution pair $(u, λ^*)$ satisfying the above equation, provided we set $u(0)=0$. Results are then extended to cover the HJB equations of mixed local-nonlocal type and this significantly improves the results in [Arapostathis-Caffarelli-Pang-Zheng (2019)].
Risk-sensitive control has received considerable interest since the seminal work of Howard and Matheson [120] because of its ability to account for fluctuations about the mean, its connection with $H_\infty$ control, and its application to financial mathematics. In this article, we attempt to put together a comprehensive survey on the research done on ergodic risk-sensitive control over the last four decades.
We study the generalized eigenvalue problem on the whole space for a class of integro-differential elliptic operators. The nonlocal operator is over a finite measure, but this has no particular structure. Some of our results even hold for singular kernels. The first part of the paper presents results concerning the existence of a principal eigenfunction. Then we present various necessary and/or sufficient conditions for the maximum principle to hold, and use these to characterize the simplicity of the principal eigenvalue.
Let $Ω$ be a bounded $C^2$ domain in $\mathbb{R}^n$ and $u\in C(\mathbb{R}^n)$ solves \begin{equation*} \begin{aligned} Δu + a Iu + C_0|Du| \geq -K\quad \text{in}\; Ω, \quad Δu + a Iu - C_0|Du|\leq K \quad \text{in}\; Ω, \quad u=0\quad \text{in}\; Ω^c, \end{aligned} \end{equation*} in the viscosity sense, where $0\leq a\leq A_0$, $C_0, K\geq 0$, and $I$ is a suitable nonlocal operator. We show that $u/δ$ is in $C^κ(\bar Ω)$ for some $κ\in (0,1)$, where $δ(x)={\rm dist}(x, Ω^c)$. Using this result, we also establish that $u\in C^{1, γ}(\barΩ)$. Finally, we apply these results to study an overdetermined problem for mixed local-nonlocal operators.