Special Session 74: Recent Advances in Local and Non-local Elliptic PDEs

Multiplicity of solutions for mixed local-nonlocal elliptic equations with singular nonlinearity

Kaushik Bal Indian Institute of Technology, Kanpur India Co-Author(s):    Stuti Das

Abstract: We prove multiplicity of solutions for the mixed local-nonlocal elliptic equation of the form $\begin{eqnarray*} \begin{split} -\Delta_pu+(-\Delta)_p^s u &= \frac{\lambda}{u^{\gamma}}+u^r \text { in } \Omega, \ u > 0 \text{ in } \Omega,\ u = 0 \text { in }\mathbb{R}^n \backslash \Omega; \end{split} \end{eqnarray*}$ where $\begin{equation*} (-\Delta )_p^s u(x)= c_{n,s}\operatorname{P.V.}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}} d y. \end{equation*}$ Under the assumptions that $\Omega$ is a smooth bounded domain in $\mathbb{R}^{n}$, $1$

Hodge decomposition in variable exponent spaces with applications to regularity theory

Anna Balci Charles University, Bielefeld University Germany Co-Author(s):

Abstract: In this talk, we explore the Hodge Laplacian in variable exponent spaces with differential forms on smooth manifolds. We present several results, including the Hodge decomposition in variable exponent spaces and a priori estimates. As an application, we derive Calderon-Zygmund estimates for variable exponent problems involving differential forms and discuss numerical approximations for nonlinear models with differential forms, which have applications in superconductivity. This presentation is based on several works with Swarnendu Sil, Michail Surnachev, and Alex Kaltenbach.

An interpolation approach to $L^{\infty}$ a priori estimates for elliptic problems with nonlinearity on the boundary

Maya Chhetri The University of North Carolina at Greensboro USA Co-Author(s):    N. Mavinga and R. Pardo

Abstract: In this talk, we present an explicit $L^{\infty}(\Omega)$ estimate for weak solutions to subcritical elliptic problems with nonlinearity on the boundary, expressed in terms of powers of their $H^1(\Omega)$ norm. Our approach relies on the already available regularity results, established using Moser's iteration technique, elliptic regularity and Gagliardo–Nirenberg interpolation inequality. We illustrate our result with an application to subcritical problems satisfying Ambrosetti-Rabinowitz condition.

Rate of Convergence of Approximations to Nonlocal HJB Equations

Indranil Chowdhury Indian Institute of Technology, Kanpur India Co-Author(s):    Espen R. Jakobsen

Abstract: We discuss monotone approximation schemes of fractional and nonlocal Hamilton-Jacobi-Bellman (HJB) equations. It is well-known in analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: Strongly degenerate problems with Lipschitz solutions, and weakly non-degenerate problems where the solutions have bounded fractional derivatives. We study the error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions.

ANISOTROPIC p-LAPLACE EQUATIONS ON LONG CYLINDRICAL DOMAIN

Purbita Jana Madras School of Economics India Co-Author(s):

Abstract: The main aim of this talk is to study the Poisson type problem for anisotropic p-Laplace type equation on long cylindrical domains. The rate of convergence is shown to be expo- nential, thereby improving earlier known results for similar type of operators. Poincare inequality for pseudo p-Laplace operator on an infinite strip-like domain is also studied and the best constant, like many other situations in literature for other operators, is shown to be the same with the best Poincare constant of an analogous problem set on a lower dimension.

Semilinear elliptic boundary value problems on the exterior of a ball in $R^{n}$, $n \\geq 2$

Lakshmi Sankar Kalappattil Indian Institute of Technology Palakkad India Co-Author(s):    Anumol Joseph

Abstract: We consider problems of the form, $\begin{equation} \begin{cases} - \Delta u &= \lambda K(x) f(u) \mbox { in } B_1^c, \ u(x)&=0 \mbox { on } \partial B_1, \ \end{cases} \end{equation}$ where $B_1 ^c = \{ x\in \mathbb{R}^n: |x|>1 \}, n \geq 2$, $\lambda$ is a positive parameter, and $K: B_1 ^c \rightarrow \mathbb{R}^{+}$, $f:(0,\infty) \rightarrow \mathbb{R}$ belong to classes of continuous functions with $K$ satisfying certain decay assumptions. For various classes of reaction terms and non radial weight functions, we will discuss the existence of positive solutions to such problems.

Optimal harvesting for a logistic model with grazing

Mohan Kumar Mallick VNIT Nagpur India Co-Author(s):    Mohan Mallick, Ardra A, and Sarath Sasi

Abstract: We consider semi-linear elliptic equations of the following form: $\begin{equation*} \left\{ \begin{aligned} -\Delta u &= \lambda[u-\dfrac{u^2}{K}-c \dfrac{u^2}{1+u^2}-h(x) u]=:\lambda f_h(u) &&\rm{in} ~ \Omega,\ \frac{\partial u}{\partial \eta}&+qu = 0 &&\rm{on} ~ \partial\Omega, \end{aligned} \right. \end{equation*}$ where, $h\in U=\{h\in L^2(\Omega): 0\leq h(x)\leq H\}.$ We prove the existence and uniqueness of the positive solution for large $\lambda.$ Further, we establish the existence of an optimal control $h\in U$ that maximizes the functional $J(h)=\int_{\Omega}h(x)u_h(x)~{\rm{d}}x-\int_{\Omega}(B_1+B_2 h(x))h(x)~{\rm{d}}x$ over $U$, where $u_h$ is the unique positive solution of the above problem associated with $h$, $B_1>0$ is the cost per unit effort when the level of effort is low and $B_2>0$ represents the rate at which the cost rises as more labor is employed. Finally, we provide a unique optimality system.

High energy solutions for non-compact variational problems

Saikat Mazumdar Indian Institute of Technology Bombay India Co-Author(s):

Abstract: In this talk, we will revisit the topological method of Coron to obtain solutions of a critical exponent polyharmonic equation. A key step is to show that in the absence of minimizers, a small perturbation of the first energy level can be embedded into the domain.

Nonlinear nonlocal potential theory at the gradient level

Simon Nowak Bielefeld University Germany Co-Author(s):    Lars Diening, Kyeongbae Kim, Ho-Sik Lee

Abstract: I will present pointwise gradient potential estimates for a class of nonlinear nonlocal equations related to quadratic nonlocal energy functionals. These pointwise estimates imply that the first-order regularity properties of such general nonlinear nonlocal equations coincide with the sharp ones of the fractional Laplacian. The talk is based on joint work with Lars Diening (Bielefeld), Kyeongbae Kim (Seoul) and Ho-Sik Lee (Bielefeld).

Asymptotic Estimates for $(p,q)$ Laplace Problems with Singular and Indefinite Sign Non-linearity and some applications

Dhanya Rajendran IISER Thiruvananthapuram India Co-Author(s):

Abstract: We will focus on the asymptotic behavior of the solutions to the boundary value problem $$ -\Delta_p u -\Delta_q u = \lambda g(x)\ \mbox{in}\ \Omega$$ and $$ u =0\ \mbox{on}\ \partial \Omega$$ as $\lambda$ approaches b $\infty$ where $\Omega$ in a smooth bounded domain in $\mathbb{R}^N$ and $g: \Omega \rightarrow \mathbb{R}$ is indefinite in sign and possibly singular near the boundary of $\Omega.$ These estimates find application in establishing the existence of a positive solution to a related problem $$ -\Delta_p u -\Delta_q u = \lambda f(u)\ \mbox{in}\ \Omega$$ with zero boundary conditions. Here we consider the non-linearity $f:(0,\infty) \rightarrow \mathbb{R}$ to be $p$-sublinear at infinity. Moreover, when $f$ takes a specific form, we obtain a positive solution that also serves as a local minimizer for the associated energy functional.

Degenerate Schr{\o}dinger-Kirchhoff $(p, N)$-Laplacian problem with singular Trudinger-Moser nonlinearity in $\mathbb{R}^N$

Abhishek Sarkar Indian Institute of Technology Jodhpur India Co-Author(s):    Deepak Kumar Mahanta, Tuhina Mukherjee

Abstract: In this talk, we will discuss the existence of nontrivial nonnegative solutions for a $(p, N)$-Laplacian Schr{\o}dinger-Kirchhoff problem in $\mathbb{R}^N$ with singular exponential nonlinearity. The main features of the work are the $(p, N)$ growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger-Moser inequality, and a completely new Br\`ezis-Lieb type lemma for singular exponential nonlinearity.

Uniqueness of positive solutions for a class of nonlinear elliptic equations with Robin boundary conditions

Ratnasingham Shivaji University of North Carolina at Greensboro USA Co-Author(s):    D.D. Hai & X.Wang

Abstract: We prove uniqueness of positive solutions to the BVP $\begin{equation*} \left\{ \begin{array}{c} -\Delta u=\lambda f(u)\ \text{\ in }\Omega , \ \frac{\partial u}{\partial n}+bu=0\ \text{\ on }\partial \Omega ,% \end{array}% \right. \end{equation*}$ when the parameter $\lambda $ is large independent of $b\in \mathbb{(}% 0,\infty )$. Here $\Omega $ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial \Omega ,\ f:[0,\infty )\rightarrow \mathbb{[}% 0,\infty )\mathbb{\ }$is continuous, sublinear at $\infty $, and satisfies a concavity-like condition for $u$ large.

On logarithmic p-Laplacian

Firoj Sk University of Oldenburg Germany Co-Author(s):    B. Dyda and S. Jarohs

Abstract: We study the logarithmic $p$-Laplacian $L_{\Delta_p}$, which arises as formal derivative of the fractional $p$-Laplacian $(-\Delta_p)^s$ at $s=0$. We present a variational framework to study the Dirichlet problems involving the $L_{\Delta_p}$ in bounded domains and use it to characterize the asymptotics of principal Dirichlet eigenvalues and eigenfunctions of $(-\Delta_p)^s$ as $s\to 0$. As a byproduct, we then derive a Faber-Krahn type inequality for the principal Dirichlet eigenvalue of $L_{\Delta_p}$. In addition, we discuss a boundary Hardy-type inequality for the spaces associated with the weak formulation of the logarithmic $p$-Laplacian. This talk is based on joint work with B. Dyda(Wroclaw) and S. Jarohs(Frankfurt).

Shape optimization problem for Steklov Dirichlet eigenvalues

Sheela Verma Indian Institute of Technology (BHU) Varanasi India Co-Author(s):    Sagar Basak, Anisa Chorwadwala

Abstract: Let $\Omega$ be a bounded smooth domain in ${R}^{n}$ with two disjoint boundary components $C_1$ and $C_2$. The mixed Steklov Dirichlet problem is to find harmonic function $u$ in $\Omega$ such that $u=0$ on $C_1$ and outer normal derivative of $u$ is directly proportional to $u$ along $C_2$. This problem models the stationary heat distribution in $\Omega$ with the conditions that the temperature along $C_1$ is kept to zero and that the heat flux through $C_2$ is proportional to the temperature. In this talk, I will first discuss about behaviour of the first Steklov Dirichlet eigenvalue on doubly connected domains and then provide some isoperimetric bounds for higher Steklov Dirichlet eigenvalues. I will also talk about similar bounds for eigenvalues of higher Steklov Neumann eigenvalues.