Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.29462

A Riemannian Extension of a Quadrature Surface Free Boundary Problem: Stability and Optimality

Ababacar Sadikhe Djite·Diaraf Seck
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study a quadrature surface free boundary problem on a smooth compact finite-dimensional Riemannian manifold $(M,g)$. The problem is formulated as a shape optimization problem involving a Dirichlet problem for the Laplace--Beltrami operator, with a geometric condition on the free boundary. Under suitable uniform geometric assumptions on the admissible class, we establish its compactness and prove the stability of the corresponding Dirichlet problems, including strong convergence of the associated states in $H_0^1(M)$. We derive the first-order optimality condition for the associated shape functional. Finally, we establish a Riemannian comparison principle for the second fundamental forms and the mean curvatures of tangent boundaries at a contact point. These results provide a rigorous extension of the quadrature surface free boundary framework from the Euclidean setting to compact Riemannian manifolds.

Explore related subjects

math.APmath.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ababacar Sadikhe Djite, Diaraf Seck. 2026-08-29. A Riemannian Extension of a Quadrature Surface Free Boundary Problem: Stability and Optimality. https://arxiv.org/abs/2608.29462

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Renormalized Lambert-W Cascade and Finite-Time Amplification and Blowup for reconstructed b Dynamics on T^3 for the 3D Navier Stokes Equations

This article extracts and consolidates the renormalized Lambert-$W$ branch-point cascade, its distinguished phase reduction, the exact characteristic invariant and finite-time amplification mechanism, and the extended reconstructed $b_i$ equation on $\mathbb T^3$. Repeated historical derivations are removed while the principal proofs and terminal reconstruction estimates are retained. The presentation separates exact finite-depth statements from coupled-depth asymptotics and records the hypotheses required for the extended PDE reconstruction. This paper further supports a recent paper \cite {moschandreou2026exploration} published by the corresponding author which claims that the Navier Stokes equations lose smoothness in finite time from initial smooth data.

math.AP↗
arXiv2026

Unconditional uniqueness for the derivative nonlinear Schrödinger equation by normal form approach

We prove uniqueness of solutions to the Cauchy problem for the derivative nonlinear Schrödinger equation in $L^\infty_tH^{1/2}_x$. Our proof is based on the method of normal form reduction (NFR), which has been employed to obtain the uniqueness in $C_tH^s_x$, $s>1/2$. To overcome logarithmic divergences at the $H^{1/2}$ regularity, we exploit the $B^{0+}_{\infty,1}$ control of solutions provided by a refined Strichartz estimate. Our NFR argument consists of two stages: we first use NFR finitely many times to derive an intermediate equation in which the main cubic nonlinearity is restricted to a certain type of frequency interaction; we then apply the infinite NFR scheme to the intermediate equation. Moreover, we modify the usual NFR argument relying on continuity in time of solutions so that the uniqueness in the class $L^\infty_tH^{1/2}_x$ can be obtained directly.

math.AP↗
arXiv2026

Equivalence between solvability of the Dirichlet and Regularity problem under an $L^1$ Carleson condition on $\partial_t A$

We study an elliptic operator $L:=-\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that solvability of the Regularity problem in $\dot{W}^{1,p}$ implies solvability of the adjoint Dirichlet problem in $L^{p'}$. Previously, Shen (2007) established a partial reverse result. In our work, we show that if we assume a mixed \(L^1-L^\infty\) condition on only \(|\partial_t A|\), the full reverse direction holds. As a result, we obtain equivalence between solvability of the Dirichlet problem $(D)^*_{p'}$ and the Regularity problem $(R)_p$ under this condition. As a further consequence, we can extend the class of operators for which the $L^p$ Regularity problem is solvable by operators satisfying the mixed $L^1-L^\infty$ condition. Additionally in the case of the upper half plane, this class includes operators satisfying this this mixed \(L^1-L^\infty\) condition.

math.AP↗