Sobolev regularity for first order mean field games

In this paper we obtain Sobolev estimates for weak solutions of first order variational Mean Field Game systems with coupling terms that are local functions of the density variable. Under some coercivity conditions on the coupling, we obtain first order Sobolev estimates for the density variable, while under similar coercivity conditions on the Hamiltonian we obtain second order Sobolev estimates for the value function. These results are valid both for stationary and time-dependent problems. In the latter case the estimates are fully global in time, thus we resolve a question which was left open in [23]. Our methods apply to a large class of Hamiltonians and coupling functions.     

Sobolev spaces of symmetric functions and applications

We prove sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball. As a consequence, we obtain some imbeddings of such Sobolev spaces in weighted Lq-spaces. We also prove similar imbeddings for Sobolev spaces of functions with partial symmetry. Our techniques lead to new Hardy type inequalities. It is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates. We apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the Hénon type under both Dirichlet and Navier boundary conditions. The delicate question of the regularity of these solutions is also established.     

A Sum Operator with Applications to Self-Improving Properties of Poincaré Inequalities in Metric Spaces

We define a class of summation operators with applications to the self-improving nature of Poincaré–Sobolev estimates, in fairly general quasimetric spaces of homogeneous type. We show that these sum operators play the familiar role of integral operators of potential type (e.g., Riesz fractional integrals) in deriving Poincaré–Sobolev estimates in cases when representations of functions by such integral operators are not readily available. In particular, we derive norm estimates for sum operators and use these estimates to obtain improved Poincaré–Sobolev results.     

Sobolev and Hardy–Littlewood–Sobolev inequalities

This paper is devoted to improvements of Sobolev and Onofri inequalities. The additional terms involve the dual counterparts, i.e. Hardy–Littlewood–Sobolev type inequalities. The Onofri inequality is achieved as a limit case of Sobolev type inequalities. Then we focus our attention on the constants in our improved Sobolev inequalities, that can be estimated by completion of the square methods. Our estimates rely on nonlinear flows and spectral problems based on a linearization around optimal Aubin–Talenti functions.     

On the boundary value problem of the biharmonic operator on domains with angular corners

The paper is concerned with boundary singularities of weak solutions of boundary value problems governed by the biharmonic operator. The presence of angular corner points or points at which the type of boundary condition changes in general causes local singularities in the solution. For that case the general theory of V. A. Kondrat'ev provides a priori estimates in weighted Sobolev norms and asymptotic singular representations for the solution which essentially depend on the zeros of certain transcendental functions. The distribution of these zeros will be analysed in detail for the biharmonic operator under several boundary conditions. This leads to sharp a priori estimates in weighted Sobolev norms where the weight function is characterized by the inner angle of the boundary corner. Such estimates for “negative” Sobolev norms are used to analyse also weakly nonlinear perturbations of the biharmonic operator as, for instance, the von Kármán model in plate bending theory and the stream function formulation of the steady state Navier-Stokes problem. It turns out that here the structure of the corner singularities is essentially the same as in the corresponding linear problem.     

Analysis of finite element methods for second order boundary value problems using mesh dependent norms

This paper presents a new approach to the analysis of finite element methods based onC ⁰-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL ₂ norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL ₂ norm and in the 2nd order Sobolev norm — the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform.     

Tangential Interpolation in Weighted Vector-valued H p Spaces, with Applications

In this paper, norm estimates are obtained for the problem of minimal-norm tangential interpolation by vector-valued analytic functions in weighted H^p spaces, expressed in terms of the Carleson constants of related scalar measures. Applications are given to the notion of p-controllability properties of linear semigroup systems and controllability by functions in certain Sobolev spaces.     

Sobolev embedding in ℂn and the\overline \partial - equation

We prove a Sobolev embedding theorem for functions that are in a Sobolev space while their is in L^t, fort large enough. This allows us to deduceL ^p or Lipschitz estimates with loss from classical Sobolev estimates for the solution of in weakly pseudo-convex domains.     

A new class of radial basis functions with compact support

Radial basis functions are well-known and successful tools for the interpolation of data in many dimensions. Several radial basis functions of compact support that give rise to nonsingular interpolation problems have been proposed, and in this paper we study a new, larger class of smooth radial functions of compact support which contains other compactly supported ones that were proposed earlier in the literature.

     

Improved interpolation inequalities,relative entropy and fast diffusion equations

We consider a family of Gagliardo–Nirenberg–Sobolev interpolation inequalities which interpolate between Sobolev?s inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions. We give an explicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants. Our approach is based on nonlinear evolution equations and improved entropy–entropy production estimates along the associated flow. Optimizing a relative entropy functional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution equations, which can be interpreted as the best fit of the solution in the asymptotic regime among all asymptotic profiles.     

Extension Theorems for Spaces Arising from Approximation by Translates of a Basic Function

We establish extension theorems for functions in spaces which arise naturally in studying interpolation by radial basic functions. These spaces are akin in some way to the non-integer-valued Sobolev spaces, although they are considerably more general. Such extensions allow us to establish local error estimates in a way which we make precise in the introductory section of our paper. There are many other applications of these fundamental results, including improved L_p error estimates for interpolation by shifts of a single basic function, but these applications have been left to a later paper.     

Pointwise behaviour of <Emphasis Type="Italic">M</Emphasis><Superscript>1,1</Superscript> Sobolev functions

Our main objective is to study Haj?asz type Sobolev functions with the exponent one on metric measure spaces equipped with a doubling measure. We show that a discrete maximal function is bounded in the Haj?asz space with the exponent one. This implies that every such function has Lebesgue points outside a set of capacity zero. We also show that every Haj?asz function coincides with a Hölder continuous Haj?asz function outside a set of small Hausdorff content. Our proofs are based on Sobolev space estimates for maximal functions.     

A posteriori error estimates with the finite element method of lines for a Sobolev equation

A posteriori error estimates for semidiscrete finite element methods for a nonlinear Sobolev equation are considered. The error estimates are obtained by solving local nonlinear or linear pseudo‐parabolic equations for corrections to the solution on each element. The ratios of these estimates and the true errors are proved to converge to 1, implying that the estimates can be used as indicators in adaptive schemes for the problem. Numerical results underline our theoretical results. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Nonlinear potential theory for Sobolev spaces on Carnot groups

Considering Bessel kernels on a Carnot group, we establish the main facts of nonlinear potential theory: a Wolff-type inequality, capacity estimates, and a strong capacity inequality. Deriving corollaries, we give an inequality of Sobolev-Adams type and relations between the capacity and Hausdorff measure, as well as lower bounds on the Teichmüller capacity. These yield the continuity of monotone functions of a Sobolev class and some estimates applicable to studying the fine properties of functions.     

Estimates of anisotropic Sobolev spaces with mixed norms for the Stokes system in a half-space

We are concerned with the non-stationary Stokes system with non-homogeneous external force and non-zero initial data in \({\mathbb {R}}^n_+ \times (0,T)\). We obtain new estimates of solutions including pressure in terms of mixed anisotropic Sobolev spaces. As an application, some anisotropic Sobolev estimates are presented for weak solutions of the Navier–Stokes equations in a half-space in dimension three.     

Graded mesh refinement and error estimates of higher order for DGFE solutions of elliptic boundary value problems in polygons

Error estimates for DGFE solutions are well investigated if one assumes that the exact solution is sufficiently regular. In this article, we consider a Dirichlet and a mixed boundary value problem for a linear elliptic equation in a polygon. It is well known that the first derivatives of the solutions develop singularities near reentrant corner points or points where the boundary conditions change. On the basis of the regularity results formulated in Sobolev–Slobodetskii spaces and weighted spaces of Kondratiev type, we prove error estimates of higher order for DGFE solutions using a suitable graded mesh refinement near boundary singular points. The main tools are as follows: regularity investigation for the exact solution relying on general results for elliptic boundary value problems, error analysis for the interpolation in Sobolev–Slobodetskii spaces, and error estimates for DGFE solutions on special graded refined meshes combined with estimates in weighted Sobolev spaces. Our main result is that there exist a local grading of the mesh and a piecewise interpolation by polynoms of higher degree such that we will get the same order O (h^α) of approximation as in the smooth case. © 2011 Wiley Periodicals, Inc. Numer Mehods Partial Differential Eq, 2012     

Galerkin methods with splines for singular integral equations over (0, 1)

Summary In this paper a convergence analysis of Galerkin methods with splines for strongly elliptic singular integral equations over the interval (0, 1) is given. As trial functions we utilize smoothest polynomial splines on arbitrary meshes and continuous splines on special nonuniform partitions, multiplied by a weight function. Using inequalities of Gårding type for singular integral operators in weightedL ² spaces and the complete asymptotics of solutions at the endpoints, we provide error estimates in certain Sobolev norms.     

Linear independence and divided derivatives of a Drinfeld module II

In this note we extend our previous results on the linear independence of values of the divided derivatives of exponential and quasi-periodic functions related to a Drinfeld module to divided derivatives of values of identity and quasi-periodic functions evaluated at the logarithm of an algebraic value. The change in point of view enables us to deal smoothly with divided derivatives of arbitrary order. Moreover we treat a full complement of quasi-periodic functions corresponding to a basis of de Rham cohomology.

     

Dynamics of Thermally-Insulated Nonequilibrium Stefan Problem

We study a two-phase Stefan problem with kinetics. Here we prove existence of a finite-dimensional attractor for the problem without heat losses. Fot the most part we use a more elegant technique of energetic type estimates in appropriately defined weighted Sobolev spaces as opposite to the parabolic potentials of [9]. We demonstrate existence of compact attractors in the Sobolev spaces and prove that the attractor consists of sufficiently regular functions. This allows us to show that the Hausdorff dimension of the attractor is finite.     

Adaptive frame methods for elliptic operator equations: the steepest descent approach

Massimo Fornasier^¶ Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università "La Sapienza" in Roma, Via Antonio Scarpa, 16/B, I-00161 Roma, Italy Rob Stevenson^|| Department of Mathematics, Utrecht University, PO Box 80.010, NL-3508 TA Utrecht, The Netherlands ^{This paper is concerned with the development of adaptive numericalmethods for elliptic operator equations. We are particularlyinterested in discretization schemes based on wavelet frames.We show that by using three basic subroutines an implementable,convergent scheme can be derived, which, moreover, has optimalcomputational complexity. The scheme is based on adaptive steepestdescent iterations. We illustrate our findings by numericalresults for the computation of solutions of the Poisson equationwith limited Sobolev smoothness on intervals in 1D and L-shapeddomains in 2D.}