On analysis of problems of mathematical physics with non-Lipschitz boundaries

2025/09/02 by Rozanova-Pierrat, Anna
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2509.02176

Abstract

We review recent advances in solving problems of mathematical physics on domains with irregular boundaries in Rn. We distinguish two frameworks: a measure-free approach in the image of the trace operator spaces for extension domains and an L2-approach depending on a d-upper regular boundary measure. In both cases, the domains can have boundaries with different Hausdorff dimensions inside the interval (n -- 2, n). The generalization of the Poincaré-Steklov/Dirichlet-to-Neumann operator for these two contexts is given. To illustrate the established convergence of spectral problems for elliptic operators with Robin boundary conditions, we give a numerical example of the stability of localized eigenfunctions, using results of M. Graffin.

Citations

• Kernels of trace operators via fine continuity
• Fractal curvatures and short-time asymptotics of heat content
• Convergence of layer potentials and Riemann-Hilbert problem on extension domains
• High-order numerical integration on self-affine sets
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• Layer potential operators for transmission problems on extension domains
• Existence of optimal shapes for heat diffusions across irregular interfaces
• Gromov-Hausdorff distance with boundary and its application to Gromov hyperbolic spaces and uniform spaces
• On the existence of optimal shapes in architecture
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• Dirichlet boundary valued problems for linear and nonlinear wave equations on arbitrary and fractal domains
• Mixed boundary valued problem for linear and nonlinear wave equations in domains with fractal boundaries
• On the viscous Burgers equation on metric graphs and fractals
• Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d -sets
• Optimal absorption of acoustical waves by a boundary
• Short time heat diffusion in compact domains with discontinuous\n transmission boundary conditions
• The Dirichlet-to-Neumann operator on rough domains
• Sectorial forms and degenerate differential operators
• Sobolev embeddings, extensions and measure density condition

Related

• Generalization of Rellich-Kondrachov theorem and trace compacteness in the framework of irregular and fractal boundaries
• Poincaré-Steklov operator and Calderón's problem on extension domains
• Embedding Theorems and Boundary-value Problems for cusp domains
• Sharp Boundary Trace Theory and Schrödinger Operators on Bounded Lipschitz Domains
• Layer potential operators for transmission problems on extension domains