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This paper is intended to give a characterization of the optimality case in Nashs inequality, based on methods of nonlinear analysis for elliptic equations and techniques of the calculus of variations. By embedding the problem into a family of Gagliardo-Nirenberg inequalities, this approach reveals why optimal functions have compact support and also why optimal constants are determined by a simple spectral problem.
We give a direct analytic proof of the classical Boundary Harnack inequality for solutions to linear uniformly elliptic equations in either divergence or non-divergence form.
Let $(mathcal{M},g_0)$ be a compact Riemannian manifold-with-boundary. We present a new proof of the classical Gaffneys inequality for differential forms in boundary value spaces over $mathcal{M}$, via the variational approach `{a} la Kozono--Yanagis
We show how Turans inequality $P_n(x)^2-P_{n-1}(x)P_{n+1}(x)geq 0$ for Legendre polynomials and related inequalities can be proven by means of a computer procedure. The use of this procedure simplifies the daily work with inequalities. For instance,
A simple proof of the weighted two variable geometric-arithmetic a mean inequality based on one given earlier valid only for integer weights
In this paper, we give a harmonic analysis proof of the Neumann boundary observability inequality for the wave equation in an arbitrary space dimension. Our proof is elementary in nature and gives a simple, explicit constant. We also extend the metho