Generalized Fredholm Properties for Invariant Pseudodifferential Operators

We define classes of pseudodifferential operators on G-bundles with compact base and give a generalized L ² Fredholm theory for invariant operators in these classes in terms of von Neumann’s G-dimension. We combine this formalism with a generalized Paley–Wiener theorem, valid for bundles with unimodular structure groups, to provide solvability criteria for invariant operators. This formalism also gives a basis for a G-index for these operators. We also define and describe a transversal dimension and its corresponding Fredholm theory in terms of anisotropic Sobolev estimates, valid also for similar bundles with nonunimodular structure group.     

Topological and analytical properties of Sobolev bundles,I: The critical case

We study the interrelationship between topological and analytical properties of Sobolev bundles and describe some of their applications to variational problems on principal bundles. We in particular show that the category of Sobolev principal G-bundles of class W ^2,m/2 defined over M ^m is equivalent to the category of smooth principal G-bundles on M and give a characterization of the weak sequential closure of smooth principal G-bundles with prescribed isomorphism class. We also prove a topological compactness result for minimizing sequences of a conformally invariant Yang-Mills functional.      

To the Sobolev embedding theorem for the limiting exponent

We establish embeddings of the Sobolev space W _p ^s and the space B _pq ^s (with the limiting exponent) in certain spaces of locally integrable functions of zero smoothness. This refines the embedding of the Sobolev space in the Lorentz and Lorentz-Zygmund spaces. Similar problems are considered for the case of irregular domains and for the potential space.     

Topological rearrangement of a function

We extend our results on weak diffeomorphism classes and decompositions of Sobolev functions to a more general framework. We introduce a family of decompositions of Sobolev functions W₀^1,p rich enough that we conjecture it allows decomposition of all Sobolev functions, not just the “craterless” ones considered in [7]. The associated weak diffeomorphism classes of a W₀^1,p Sobolev function are weakly closed when p ≥ n.     

Existence and uniqueness of solution in Sobolev space for an unsteady crystal growth problem with zero surface tension

We study an initial value problem for a two-dimensional dendritic crystal growth model with zero surface tension. If the initial data is in Sobolev space H²(R), it is proved that an unique local solution exists in proper Sobolev space.     

A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems

We make a contribution to the theory of embeddings of anisotropic Sobolev spaces into L ^p -spaces (Sobolev case) and spaces of Hölder continuous functions (Morrey case). In the case of bounded domains the generalized embedding theorems published so far pose quite restrictive conditions on the domain’s geometry (in fact, the domain must be “almost rectangular”). Motivated by the study of some evolutionary PDEs, we introduce the so-called “semirectangular setting”, where the geometry of the domain is compatible with the vector of integrability exponents of the various partial derivatives, and show that the validity of the embedding theorems can be extended to this case. Second, we discuss the a priori integrability requirement of the Sobolev anisotropic embedding theorem and show that under a purely algebraic condition on the vector of exponents, this requirement can be weakened. Lastly, we present a counterexample showing that for domains with general shapes the embeddings indeed do not hold.     

A proof of Hessian Sobolev inequality

In this paper, taking the Hessian Sobolev inequality (0<p≤k) (X.-J. Wang, 1994 [2]) as the starting point, we give a proof of the Hessian Sobolev inequality when k<p≤k^∗, where k^∗ is the critical Sobolev embedding index of k-Hessian type. We also prove that k^∗ is optimal by one-dimensional Hardy’s inequality.     

The Gradient Flow of the L 2 Curvature Functional with Small Initial Energy

We investigate the low-energy behavior of the gradient flow of the L ² norm of the Riemannian curvature on a four-manifold. In particular we show that if the initial energy is chosen small enough with respect to the initial Sobolev constant and the H ¹ norm of the gradient vector then the flow exists for all time and converges to a flat metric. We also improve the regularity requirement for the flow proved in Streets (J. Geom. Anal. 18:249, 2008) in the case of four-manifolds.     

To the Theory of Sobolev-Type Classes of Functions with Values in a Metric Space

We study some classes of functions with values in a complete metric space which can be considered as analogs of the Sobolev spaces W _p ¹ . Earlier the author considered the case of functions on a domain of ? ⁿ . Here we study the general case of mappings on an arbitrary Lipschitz manifold. We give necessary auxiliary facts, consider some examples, and describe some methods of construction of lower semicontinuous functionals on the classes W _p ¹ (M), where M is a Lipschitz manifold.     

A Hardy inequality in the half-space

Here we prove a Hardy-type inequality in the upper half-space which generalize an inequality originally proved by Maz’ya (Sobolev Spaces, Springer, Berlin, 1985, p. 99). Here we present a different proof, which enable us to improve the constant in front of the remainder term. We will also generalize the inequality to the L^p case.     

Strong solutions to the Richards equation in the unsaturated zone

We study the unsaturated case of the Richards equation in three space dimensions with Dirichlet boundary data. We first establish an a priori L^∞-estimate. With its help, by means of a fixed point argument we prove global in time existence of a unique weak solution in Sobolev spaces. Finally, we are able to improve the regularity of this weak solution in order to gain a strong one.     

In Metric-measure Spaces Sobolev Embedding is Equivalent to a Lower Bound for the Measure

We study Sobolev inequalities on doubling metric measure spaces. We investigate the relation between Sobolev embeddings and lower bound for measure. In particular, we prove that if the Sobolev inequality holds, then the measure μ satisfies the lower bound, i.e. there exists b such that μ(B(x,r))≥b r ^α for r∈(0,1] and any point x from metric space.     

Another note on the embedding of the Sobolev space for the limiting exponent

The embedding of the Sobolev spaces W _p ^s (? ⁿ ) in a Lizorkin-type space of locally summable functions of zero smoothness is established. This result is extended to the case of the embedding of Sobolev spaces on nonregular domains of n-dimensional Euclidean space. The formulation of the theorem depends on the geometric parameters of the domain of the functions.     

Approximation by polynomials and smooth functions in Sobolev spaces with respect to measures

The density of polynomials is straightforward to prove in Sobolev spaces W^k,p((a,b)), but there exist only partial results in weighted Sobolev spaces; here we improve some of these theorems. The situation is more complicated in infinite intervals, even for weighted L^p spaces; besides, in the present paper we have proved some other results for weighted Sobolev spaces in infinite intervals.     

Embedding of Sobolev spaces and properties of the domain

We establish the embedding of the Sobolev space W _p ^s (G) ? L _q (G) for an irregular domain G in the case of a limit exponent under new relations between the parameters depending on the geometric properties of the domain G.     

On a parabolic logarithmic Sobolev inequality

In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi [H. Kozono, Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000) 191-200] have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) BMO norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of anisotropic (parabolic) BMO norm. More precisely we give an upper bound for the L^∞ norm of a function in terms of its parabolic BMO norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long-time existence result for a class of nonlinear parabolic problems.     

Fonctions lipschitziennes et espaces de Sobolev fractionnaires

We consider a semigroup of Markovian and symmetric operators to which we associate fractional Sobolev spaces D^α_p (0 < α < 1 and 1 < p < ∞) defined as domains of fractional powers (−A_p)^α/2, where A_p is the generator of the semigroup in L^p. We show under rather general assumptions that Lipschitz continuous functions operate by composition on D^α_p if p ≥ 2. This holds in particular in the case of the Ornstein-Uhlenbeck semigroup on an abstract Wiener space.     

Traces of Sobolev functions on the Ahlfors sets of Carnot groups

We prove the converse of the trace theorem for the functions of the Sobolev spaces W _p ^l on a Carnot group on the regular closed subsets called Ahlfors d-sets (the direct trace theorem was obtained in one of our previous publications). The theorem generalizes Johnsson and Wallin’s results for Sobolev functions on the Euclidean space. As a consequence we give a theorem on the boundary values of Sobolev functions on a domain with smooth boundary in a two-step Carnot group. We consider an example of application of the theorems to solvability of the boundary value problem for one partial differential equation.     

Sobolev problem in the complete scale of Banach Spaces

In a bounded domainG ? ? ⁿ , whose boundary is the union of manifolds of different dimensions, we study the Sobolev problem for a properly elliptic expression of order 2m. The boundary conditions are given by linear differential expressions on manifolds of different dimensions. We study the Sobolev problem in the complete scale of Banach spaces. For this problem, we prove the theorem on a complete set of isomorphisms and indicate its applications.     

A nonautonomous chain rule in W 1,p and BV

In this paper we consider the chain rule formula for compositions ${x\mapsto F(x, u(x))}$ in the case when u has a Sobolev or BV regularity and F(x, z) is separately Sobolev, or BV, with respect to x and C ¹ with respect to z. Our results extend to this “nonautonomous” case the results known for compositions ${x\mapsto F(u(x))}$ .