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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

4 votes
0 answers
134 views

Eigenvalues of Schrödinger operator with Robin condition on the boundary

Let $(M^2,g)$ be a compact Riemannian surface with boundary and let $L = \Delta_g + q$ be a Schrödinger operator, where $\Delta_g = -\operatorname{div} \nabla$ is the Laplacian with respect to the met …
Eduardo Longa's user avatar
7 votes
0 answers
122 views

Steklov eigenvalue for circle valued functions

Let $(M,g)$ be a compact Riemannian manifold with boundary. It is well known that the first positive Steklov eigenvalue $\sigma_1$ of $M$ has the following variational characterization: $$\sigma_1(M,g …
Eduardo Longa's user avatar
3 votes
0 answers
78 views

Dirichlet-to-Neumann map is analytic

Let $M^n$, $n \geq 2$, be a compact smooth manifold with boundary and let $I \ni t \mapsto g_t$ be an analytic (with respect to t) $1$-parameter family of Riemannian metrics on $M$. For each $t \in I$ …
Eduardo Longa's user avatar
0 votes
0 answers
123 views

Is every minimal graph smooth?

The following result was taken from the book of Gilbarg-Trudinger: In particular, if the graph is minimal, then $u$ is smooth. Now comes my question: does the same conclusion hold for graphs over dom …
Eduardo Longa's user avatar