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Register a new userExtremizing Temperature Functions of Rods with Robin Boundary Conditions
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We compare the solutions of two one-dimensional Poisson problems on an interval with Robin boundary conditions, one with given data, and one where the data has been symmetrized. When the Robin parameter is positive and the symmetrization is symmetric decreasing rearrangement, we prove that the solution to the symmetrized problem has larger increasing convex means. When the Robin parameter equals zero (so that we have Neumann boundary conditions) and the symmetrization is decreasing rearrangement, we similarly show that the solution to the symmetrized problem has larger convex means.
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In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals $((p-1)/p)^p$ whenever Dirichlet boundary conditions are imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains.
We investigate existence and nonexistence of stationary stable nonconstant solutions, i.e. patterns, of semilinear parabolic problems in bounded domains of Riemannian manifolds satisfying Robin boundary conditions. These problems arise in several models in applications, in particular in Mathematical Biology. We point out the role both of the nonlinearity and of geometric objects such as the Ricci curvature of the manifold, the second fundamental form of the boundary of the domain and its mean curvature. Special attention is devoted to surfaces of revolution and to spherically symmetric manifolds, where we prove refined results.
Let $Omegasubsetmathbb{R}^ u$, $ uge 2$, be a $C^{1,1}$ domain whose boundary $partialOmega$ is either compact or behaves suitably at infinity. For $pin(1,infty)$ and $alpha>0$, define [ Lambda(Omega,p,alpha):=inf_{substack{uin W^{1,p}(Omega) u otequiv 0}}dfrac{displaystyle int_Omega | abla u|^p mathrm{d} x - alphadisplaystyleint_{partialOmega} |u|^pmathrm{d}sigma}{displaystyleint_Omega |u|^pmathrm{d} x}, ] where $mathrm{d}sigma$ is the surface measure on $partialOmega$. We show the asymptotics [ Lambda(Omega,p,alpha)=-(p-1)alpha^{frac{p}{p-1}} - ( u-1)H_mathrm{max}, alpha + o(alpha), quad alphato+infty, ] where $H_mathrm{max}$ is the maximum mean curvature of $partialOmega$. The asymptotic behavior of the associated minimizers is discussed as well. The estimate is then applied to the study of the best constant in a boundary trace theorem for expanding domains, to the norm estimate for extension operators and to related isoperimetric inequalities.
We classify positive solutions to a class of quasilinear equations with Neumann or Robin boundary conditions in convex domains. Our main tool is an integral formula involving the trace of some relevant quantities for the problem. Under a suitable condition on the nonlinearity, a relevant consequence of our results is that we can extend to weak solutions a celebrated result obtained for stable solutions by Casten and Holland and by Matano.
The worldline formalism has been widely used to compute physical quantities in quantum field theory. However, applications of this formalism to quantum fields in the presence of boundaries have been studied only recently. In this article we show how to compute in the worldline approach the heat kernel expansion for a scalar field with boundary conditions of Robin type. In order to describe how this mechanism works, we compute the contributions due to the boundary conditions to the coefficients A_1, A_{3/2} and A_2 of the heat kernel expansion of a scalar field on the positive real line.
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