Search results for " PD"
showing 10 items of 651 documents
In between the inequalities of Sobolev and Hardy
2015
We establish both sufficient and necessary conditions for the validity of the so-called Hardy-Sobolev inequalities on open sets of the Euclidean space. These inequalities form a natural interpolating scale between the (weighted) Sobolev inequalities and the (weighted) Hardy inequalities. The Assouad dimension of the complement of the open set turns out to play an important role in both sufficient and necessary conditions.
On the interior regularity of weak solutions to the 2-D incompressible Euler equations
2016
We study whether some of the non-physical properties observed for weak solutions of the incompressible Euler equations can be ruled out by studying the vorticity formulation. Our main contribution is in developing an interior regularity method in the spirit of De Giorgi–Nash–Moser, showing that local weak solutions are exponentially integrable, uniformly in time, under minimal integrability conditions. This is a Serrin-type interior regularity result $$\begin{aligned} u \in L_\mathrm{loc}^{2+\varepsilon }(\Omega _T) \implies \mathrm{local\ regularity} \end{aligned}$$ for weak solutions in the energy space $$L_t^\infty L_x^2$$ , satisfying appropriate vorticity estimates. We also obtain impr…
New degeneration of Fay's identity and its application to integrable systems
2011
In this paper, we find a new degenerated version of Fay's trisecant identity; this degeneration corresponds to the limit when the four points entering the trisecant identity coincide pairwise. This degenerated version of Fay's identity is used to construct algebro-geometric solutions to the multi-component nonlinear Schrodinger equation. This identity also leads to an independent derivation of algebro-geometric solutions to the Davey–Stewartson equations previously obtained in [17] in the framework of the Krichever scheme. We also give the condition of smoothness of the obtained solutions.
Evolution problems of Leray-Lions type with nonhomogeneous Neumann boundary conditions in metric random walk spaces
2019
Abstract In this paper we study evolution problems of Leray–Lions type with nonhomogeneous Neumann boundary conditions in the framework of metric random walk spaces. This covers cases with the p -Laplacian operator in weighted discrete graphs and nonlocal operators with nonsingular kernel in R N .
On $L^p$ resolvent estimates for Laplace-Beltrami operators on compact manifolds
2011
Abstract. In this article we prove Lp estimates for resolvents of Laplace–Beltrami operators on compact Riemannian manifolds, generalizing results of Kenig, Ruiz and Sogge (1987) in the Euclidean case and Shen (2001) for the torus. We follow Sogge (1988) and construct Hadamard's parametrix, then use classical boundedness results on integral operators with oscillatory kernels related to the Carleson and Sjölin condition. Our initial motivation was to obtain Lp Carleman estimates with limiting Carleman weights generalizing those of Jerison and Kenig (1985); we illustrate the pertinence of Lp resolvent estimates by showing the relation with Carleman estimates. Such estimates are useful in the …
Dirichlet approximation and universal Dirichlet series
2016
We characterize the uniform limits of Dirichlet polynomials on a right half plane. In the Dirichlet setting, we find approximation results, with respect to the Euclidean distance and {to} the chordal one as well, analogous to classical results of Runge, Mergelyan and Vitushkin. We also strengthen the notion of universal Dirichlet series.
Hitchhiker's guide to the fractional Sobolev spaces
2012
AbstractThis paper deals with the fractional Sobolev spaces Ws,p. We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results.Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.
A Density Result for Homogeneous Sobolev Spaces on Planar Domains
2018
We show that in a bounded simply connected planar domain $\Omega$ the smooth Sobolev functions $W^{k,\infty}(\Omega)\cap C^\infty(\Omega)$ are dense in the homogeneous Sobolev spaces $L^{k,p}(\Omega)$.
Nonlinear scalar field equations with general nonlinearity
2018
Consider the nonlinear scalar field equation \begin{equation} \label{a1} -\Delta{u}= f(u)\quad\text{in}~\mathbb{R}^N,\qquad u\in H^1(\mathbb{R}^N), \end{equation} where $N\geq3$ and $f$ satisfies the general Berestycki-Lions conditions. We are interested in the existence of positive ground states, of nonradial solutions and in the multiplicity of radial and nonradial solutions. Very recently Mederski [30] made a major advance in that direction through the development, in an abstract setting, of a new critical point theory for constrained functionals. In this paper we propose an alternative, more elementary approach, which permits to recover Mederski's results on the scalar field equation. T…
A quantitative isoperimetric inequality for fractional perimeters
2011
Abstract Recently Frank and Seiringer have shown an isoperimetric inequality for nonlocal perimeter functionals arising from Sobolev seminorms of fractional order. This isoperimetric inequality is improved here in a quantitative form.