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The asymptotic behaviour of surfaces with prescribed mean curvature near meeting points of fixed and free boundaries

Frank Müller (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We study the shape of stationary surfaces with prescribed mean curvature in the Euclidean 3-space near boundary points where Plateau boundaries meet free boundaries. In deriving asymptotic expansions at such points, we generalize known results about minimal surfaces due to G. Dziuk. The main difficulties arise from the fact that, contrary to minimal surfaces, surfaces with prescribed mean curvature do not meet the support manifold perpendicularly along their free boundary, in general.

The BV solution of the parabolic equation with degeneracy on the boundary

Huashui Zhan, Shuping Chen (2016)

Open Mathematics

Consider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.

The relation between the porous medium and the eikonal equations in several space dimensions.

Pierre-Louis Lions, Panagiotis E. Souganidis, Juan Luis Vázquez (1987)

Revista Matemática Iberoamericana

We study the relation between the porous medium equation ut = Δ(um), m &gt; 1, and the eikonal equation vt = |Dv|2. Under quite general assumtions, we prove that the pressure and the interface of the solution of the Cauchy problem for the porous medium equation converge as m↓1 to the viscosity solution and the interface of the Cauchy problem for the eikonal equation. We also address the same questions for the case of the Dirichlet boundary value problem.

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