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Search: MSC category 35J25 ( Boundary value problems for second-order elliptic equations )

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1. CJM 2012 (vol 65 pp. 927)

Wang, Liping; Zhao, Chunyi
Infinitely Many Solutions for the Prescribed Boundary Mean Curvature Problem in $\mathbb B^N$
We consider the following prescribed boundary mean curvature problem in $ \mathbb B^N$ with the Euclidean metric: \[ \begin{cases} \displaystyle -\Delta u =0,\quad u\gt 0 &\text{in }\mathbb B^N, \\[2ex] \displaystyle \frac{\partial u}{\partial\nu} + \frac{N-2}{2} u =\frac{N-2}{2} \widetilde K(x) u^{2^\#-1} \quad & \text{on }\mathbb S^{N-1}, \end{cases} \] where $\widetilde K(x)$ is positive and rotationally symmetric on $\mathbb S^{N-1}, 2^\#=\frac{2(N-1)}{N-2}$. We show that if $\widetilde K(x)$ has a local maximum point, then the above problem has infinitely many positive solutions that are not rotationally symmetric on $\mathbb S^{N-1}$.

Keywords:infinitely many solutions, prescribed boundary mean curvature, variational reduction
Categories:35J25, 35J65, 35J67

2. CJM 2012 (vol 65 pp. 702)

Taylor, Michael
Regularity of Standing Waves on Lipschitz Domains
We analyze the regularity of standing wave solutions to nonlinear Schrödinger equations of power type on bounded domains, concentrating on Lipschitz domains. We establish optimal regularity results in this setting, in Besov spaces and in Hölder spaces.

Keywords:standing waves, elliptic regularity, Lipschitz domain
Categories:35J25, 35J65

3. CJM 2010 (vol 62 pp. 808)

Legendre, Eveline
Extrema of Low Eigenvalues of the Dirichlet-Neumann Laplacian on a Disk
We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet--Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact $1$-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.

Keywords: Laplacian, eigenvalues, Dirichlet-Neumann mixed boundary condition, Zaremba's problem
Categories:35J25, 35P15

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