Special Session 65: Geometry of PDEs on Manifolds and Nilpotent Lie Groups

Varadhan-type asymptotics for boundary data with localized support

Diego Berti Univeristy of Turin Italy Co-Author(s):

Abstract: We study short-time and small-diffusion asymptotics for linear, uniformly elliptic operators with variable coefficients in a bounded domain. For nonconstant boundary data, we show that the leading behavior is governed by the intrinsic distance, induced by the operator, to the support of the boundary datum. This extends the classical Varadhan result, originally obtained for constant boundary data, and emphasizes the role of the localized source. This is joint work with Michele Marini (University del Sannio, Italy) and Rolando Magnanini (University di Firenze, Italy).

Qualitative properties of solutions to fractional elliptic and parabolic equations

Wenxiong Chen Yeshiva University USA Co-Author(s):

Abstract: In this talk, we will present some of the recent developments in the study of qualitative properties of solutions to various fractional elliptic and parabolic equations $$ {\cal L} u = f(t, u(x,t)), $$ where ${\cal L}$ is a fractional elliptic or parabolic operator assuming one of the following forms $$ (\Delta)^s, \;\; \partial_t + (-\Delta)^s, \;\; \partial_t^\alpha + (-\Delta)^s, \;\; (\partial_t -\Delta)^s.$$ We will illustrate the extent of non-locality of these operators and explain the differences among them. Then we will present some of our recent results on qualitative properties of solutions including monotonicity, symmetry, uniqueness, nonexistence, and a priori estimates.

Symmetry and rigidity results for Serrin`s overdetermined type problems in weighted Riemannian manifolds

Giulio Ciraolo University of Milan Italy Co-Author(s):    Laura Accornero

Abstract: We consider the classical Serrin's overdetermined boundary value problem in a weighted Riemannian manifold. Under suitable assumptions, we prove that the existence of a solution implies that the domain must be isometric to an Euclidean ball, also providing a rigidity result on the underlying manifold.

Rigidity results for spherical, as well as annular domains, in manifolds with pole

Antonio Greco University of Cagliari Italy Co-Author(s):    Marcello Lucia (New York), Pieralberto Sicbaldi (Granada)

Abstract: Overdetermined problems for the rotationally invariant Poisson equation $-\Delta u = f(r)$ in a Riemannian manifold with pole~$O$ have been recently investigated (arXiv:2602.18289) by Marcello Lucia (New York), Pieralberto Sicbaldi (Granada) and the speaker. We give conditions on~$f$ and on the boundary data implying that the solution~$u$ is radial and the domain of the problem is a geodesic ball centered at~$O$, or an annulus centered there. Our results hold, in particular, in the three space forms with constant curvature. Proofs are based on the comparison principle.

Huber`s Theorems in Dimensions 2 and 4

Paul Laurain University Gustave Eiffel France Co-Author(s):    Dorian Martino

Abstract: After recalling the classical statement of Huber`s second theorem, I will propose an approach via the moving frame method. I will then present the state of the art on generalizations of this theorem to dimension 4, emphasizing that these do not yet yield a complete generalization. Next, drawing inspiration from the moving frame approach, we will present a new advance toward a full characterization. Finally, I will give some applications to the extrinsic case. Joint work with Dorian Martino.

Concentration phenomena for Bernoulli`s free boundary problem

Michiaki Onodera Institute of Science Tokyo Japan Co-Author(s):

Abstract: Bernoulli`s free boundary problem is an overdetermined boundary value problem in which one seeks an annular domain with a fixed exterior boundary and an interior free boundary, such that the associated capacitary potential also satisfies a prescribed Neumann boundary condition on the free boundary. I will talk about recent developments regarding to a conjecture of Flucher and Rumpf that asserts the existence of a family of free boundaries concentrating at non-degenerate local minima of the Robin function.

Symmetry results in Heisenberg group, Carnot group and Grushin spaces

Jyotshana Prajapat University of Mumbai India Co-Author(s):    Anoop S. Varghese

Abstract: I will present some of the recent results related to symmetry of solutions of integral equations in the Heisenberg group and Carnot group using the moving plane method with application to the CR Yamabe problem. I will also discuss related progress for the Grushin space.

Some results about the capillary overdetermined problem

Pieralberto Sicbaldi Universidad de Granada Spain Co-Author(s):    Yuanyuan Lian

Abstract: In this talk we will discuss some results for bounded positive solutions of the capillary overdetermined problem: \[ \left\{\begin{array} {ll} \mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + f(u) = 0 & \mbox{in }\; \Omega \subset \mathbb{R}^n,\ u= 0 & \mbox{on }\; \partial \Omega, \ \partial_{\nu} u= const. &\mbox{on }\; \partial \Omega. \end{array}\right. \] In particular, we will show rigidity for some class of domains $\Omega$, including some gradient estimates, and existence of bifurcation for other classes of domains. These are joint works with Y. Lian.

Asymptotic behavior of least energy solutions to the Finsler Lane-Emden problem with large exponents

Futoshi Takahashi Osaka Metropolitan University Japan Co-Author(s):    Sadaf Habibi

Abstract: In this talk we are concerned with the least energy solutions to the Finsler Lane-Emden problem with large exponent $p$ in the nonlinearity on an $N$ dimensional bounded domain. We prove that the least energy solution is bounded from above and below independent of $p$ large. Precise asymptotic behaviors of energy and mass are obtained. We also obtain the asymptotic behavior of the sup-norm of the least energy solutions as $p$ gets large. Furthermore, we show some concentration phenomena for the scaled functions and prove that the single-point blowup cannot occur on the boundary of the domain. This talk is based on the joint work with Sadaf Habibi (DCDS, 42, no.10 (2022)).