The present note contains a review of $p$-energies and Sobolev spaces on metric measure spaces that carry a strongly local regular Dirichlet form. These Sobolev spaces are then used to generalize some basic results from the calculus of variations, such as the existence of minimizers for convex functionals and certain constrained mimimization problems. This applies to a number of non-classical situations such as degenerate diffusions, superpositions of diffusions and diffusions on fractals or on products of fractals.
We establish sharp $L^2$-Sobolev estimates for classes of pseudodifferential operators with singular symbols whose non-pseudodifferential (Fourier integral operator) parts exhibit two-sided fold singularities. The operators considered include both singular integral operators along curves in $R^2$ with simple inflection points and normal operators arising in linearized seismic imaging in the presence of fold caustics.
We investigate some multiplication properties of Kato-Sobolev spaces by adapting the techniques used in the study of Beurling algebras by Coifman and Meyer. We develop an analytic functional calculus for Kato-Sobolev algebras based on an integral representation formula belonging A. P. Calderón. Also we study the Schatten-von Neumann properties of pseudo-differential operators with symbols in the Kato-Sobolev spaces.
We prove an estimation of the Kolmogorov $ε$-entropy in H of the unitary ball in the space V, where H is a Hilbert space and V is a Sobolev-like subspace of H. Then, by means of Zelik's result [5], an estimate of the fractal dimension of the attractors of some nonlinear parabolic equations is established.
For fractional derivatives and time-fractional differential equations, we construct a framework on the basis of the operator theory in fractional Sobolev spaces. Our framework provides a feasible extension of the classical Caputo and the Riemann-Liouville derivatives within Sobolev spaces of fractional orders including negative ones. Our approach enables a unified treatment for fractional calculus and time-fractional differential equations. We formulate initial value problems for fractional ordinary differential equations and initial boundary value problems for fractional partial differential …
With a view towards Riemannian or sub-Riemannian manifolds, RCD metric spaces and specially fractals, this paper makes a step further in the development of a theory of heat semigroup based $(1,p)$ Sobolev spaces in the general framework of Dirichlet spaces. Under suitable assumptions that are verified in a variety of settings, the tools developed by D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste in the paper "Sobolev inequalities in disguise" allow us to obtain the whole family of Gagliardo-Nirenberg and Trudinger-Moser inequalities with optimal exponents. The latter depend not only on th…
We study a nonlocal Venttsel' problem in a non-convex bounded domain with a Koch-type boundary. Regularity results of the strict solution are proved in weighted Sobolev spaces. The numerical approximation of the problem is carried out and optimal a priori error estimates are obtained.
This book provides a gentle introduction to fractional Sobolev spaces, which play a central role in the calculus of variations, partial differential equations, and harmonic analysis. The first part deals with fractional Sobolev spaces of one variable. It covers the definition, standard properties, extensions, embeddings, Hardy inequalities, and interpolation inequalities. The second part deals with fractional Sobolev spaces of several variables. The author studies completeness, density, homogeneous fractional Sobolev spaces, embeddings, necessary and sufficient conditions for extensions, Gagli…
This article presents a new approach of semigroup analysis and pseudo-differential calculus for deriving the regularizing estimate on non-cutoff linearized Boltzmann equation. We are able to obtain regularizing estimate of semigroup $e^{tB}$ that is continuous from weighted Sobolev space $H(a^{-1/2})H^m_x$ to $H(a^{1/2})H^m_x$ with a sharp large time decay. With these properties, we prove the existence of global-in-time unique solution to the non-cutoff Boltzmann equation for hard potential on the whole space with weak regularity assumption on initial data. We consider the hard potential case …
We extend the Global Compactness result by M. Struwe (Math. Z, 1984) to any fractional Sobolev spaces $\dot{H}^s(Ω)$ for $0<s<N/2$ and $Ω\subset \mathbb{R}^N$ a bounded domain with smooth boundary. The proof is a simple direct consequence of the so-called Profile Decomposition of P. Gerard (ESAIM: Control, Optimisation and Calculus of Variations, 1998).
Based on recent developments in the theory of fractional Sobolev spaces, an interesting new class of nonlocal variational problems has emerged in the literature. These problems, which are the focus of this work, involve integral functionals that depend on Riesz fractional gradients instead of ordinary gradients and are considered subject to a complementary-value condition. With the goal of establishing a comprehensive existence theory, we provide a full characterization for the weak lower semicontinuity of these functionals under suitable growth assumptions on the integrands. In doing so, we s…
Sobolev spaces are a natural framework for the analysis of problems in partial differential equations and calculus of variations. Some physical and geometric contexts, such as liquid crystals models and harmonic maps, lead to consider Sobolev maps, that is, Sobolev vector functions whose range is constrained in a surface or submanifold of the space. This additional nonlinear constraint provokes the appearance of finite-energy topological singularities. These singularities are characterized by a nontrivial topological invariant such as the topological degree, they represent an obstruction to th…
This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new …
We prove a one-parameter family of diffusion hypercontractivity and present the associated Log-Sobolev, Poincare and Talagrand inequalities. A mean-field type Bakry-Emery iterative calculus and volume measure based integration formula (Yano's formula) are presented. Our results are based on the interpolation among divergence functional, generalized diffusion process, and generalized optimal transport metric. As a result, an inequality among Pearson divergence (P), negative Sobolev metric H^-1 and generalized Fisher information functional (I), named PH^{-1}I inequality, is derived.
