相关论文: An Isoperimetric Inequality for Fundamental Tones …

Consider the parabolic free boundary problem $$ \Delta u - \partial_t u = 0 \textrm{in} \{u>0\}, |\nabla u|=1 \textrm{on} \partial\{u>0\} . $$ For a realistic class of solutions, containing for example {\em all} limits of the singular…

This is a continuation of the paper 'Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes' by S. Chanillo, D. Grieser, M. Imai, K. Kurata, and I. Ohnishi. Again, we consider the following…

It is well known that minimizers of the Allen-Cahn-type functional \[ J_\epsilon(u):=\int_\Omega\frac{\epsilon|\nabla u|^2}{2}+\frac{W(u)}{\epsilon}, \] where $W$ is a double-well potential, resemble minimal surfaces in the sense that their…

We study spectral properties of Dirac operators on bounded domains $\Omega \subset \mathbb{R}^3$ with boundary conditions of electrostatic and Lorentz scalar type and which depend on a parameter $\tau\in\mathbb{R}$; the case $\tau = 0$…

An inequality for the reverse Bossel-Daners inequality is derived by means of the harmonic transplantation and the first shape derivative. This method is then applied to elliptic boundary value problems with inhomogeneous Neumann…

We explore a connection between the Finslerian area functional based on the Busemann-Hausdorff-volume form, and well-investigated Cartan functionals to solve Plateau's problem in Finsler 3-space, and prove higher regularity of solutions.…

An isoperimetric inequality for lower order nonzero Neumann eigenvalues of the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic space has been proven in this paper. About this conclusion, we would like to point out…

It is known that any subharmonic quadrature domain in two dimensions satisfies a natural inner ball condition, in other words there is a specific upper bound on the curvature of the boundary. This result directly applies to free boundaries…

Let $\Sigma$ be a $k$-dimensional complete proper minimal submanifold in the Poincar\'{e} ball model $B^n$ of hyperbolic geometry. If we consider $\Sigma$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean…

We address extremum problems for spectral quantities associated with operators of the form $\Delta^2-\tau\Delta$ with Dirichlet boundary conditions, for non-negative values of $\tau$. The focus is on two shape optimisation problems:…

In the 19th century, Lord Rayleigh conjectured that among all clamped plates with given area, the disk minimizes the fundamental tone. In the 1990s, N. S. Nadirashvili proved the conjecture in $\mathbb{R}^2$ and M. S. Ashbaugh und R. D.…

In this article, we establish a relationship between geometric quantities of a hypersurface restricted to its boundary, and the geometric quantities of its boundary as a hypersurface of the boundary of the ball. As a first application, we…

In this paper, the spectrum of the following fourth order problem \begin{equation*} \begin{cases} \Delta^2 u+\nu u=-\lambda \Delta u &\text{in } D_1,\newline u=\partial_r u= 0 &\text{on } \partial D_1, \end{cases} \end{equation*} where…

In this article, we investigate the centered isoperimetric inequality on Cartan-Hadamard manifolds endowed with a warped product structure, namely, among all bounded measurable sets of finite perimeter and prescribed volume, the geodesic…

We consider an eigenvalue problem for the biharmonic operator with Steklov-type boundary conditions. We obtain it as a limiting Neumann problem for the biharmonic operator in a process of mass concentration at the boundary. We study the…

In the present paper, we study sharp isoperimetric inequalities for the first Steklov eigenvalue $\sigma_1$ on surfaces with fixed genus and large number $k$ of boundary components. We show that as $k\to \infty$ the free boundary minimal…

We prove that the Weinstock inequality for the first nonzero Steklov eigenvalue holds in $\mathbb{R}^n$, for $n\ge 3$, in the class of convex sets with prescribed surface area. The key result is a sharp isoperimetric inequality involving…

In this note, we use an epiperimetric inequality approach to study the regularity of the free boundary for the parabolic Signorini problem. We show that if the "vanishing order" of a solution at a free boundary point is close to $3/2$ or an…

Using the Fourier transform, we obtain upper bounds for sums of eigenvalues of the free plate.

We obtained estimates for first eigenvalues of the Dirichlet boundary value problem for elliptic operators in divergence form (i.e. for the principal frequency of non-homogeneous membranes) in bounded domains $\Omega \subset \mathbb C$…