On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains

We construct a bounded C¹ domain Ω in for which the regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists f in such that the solution of in Ω and either on or on is contained in but not in for any . An analogous result holds for Sobolev spaces with .     

Non‐autonomous forms and invariance

We generalize the Beurling–Deny–Ouhabaz criterion for parabolic evolution equations governed by forms to the non‐autonomous, non‐homogeneous and semilinear case. Let be Hilbert spaces such that V is continuously and densely embedded in H and let be the operator associated with a bounded H‐elliptic form for all . Suppose is closed and convex and the orthogonal projection onto . Given and , we investigate when the solution of the non‐autonomous evolutionary problem remains in and show that this is the case if for a.e. . Moreover, we examine necessity of this condition and apply this result to a semilinear problem.     

A note on the eigenvalues of p‐fractional Hardy–Sobolev operator with indefinite weight

In this article, we study the eigenvalues of p‐fractional Hardy operator where , , , and Ω is an unbounded domain in with Lipschitz boundary containing 0. The weight function V may change sign and may have singular points. We also show that the least positive eigenvalue is simple and it is uniquely associated to a nonnegative eigenfunction. Moreover, we proved that there exists a sequence of eigenvalues as .     

Localized peaking regimes for quasilinear parabolic equations

This paper deals with the asymptotic behavior as of all weak (energy) solutions of a class of equations with the following model representative: with prescribed global energy function Here , , , Ω is a bounded smooth domain, . Particularly, in the case it is proved that the solution u remains uniformly bounded as in an arbitrary subdomain and the sharp upper estimate of when has been obtained depending on and . In the case for all , sharp sufficient conditions on degeneration of near that guarantee the above mentioned boundedness for an arbitrary (even large) solution have been found and the sharp upper estimate of a final profile of the solution when has been obtained.     

Multiple solutions for a class of quasilinear Schrödinger equations

In this paper, we study the following quasilinear Schrödinger equations of the form where , , . Some existence results for positive solutions, negative solutions and sequence of high energy solutions are obtained via a perturbation method.     

A Szegő limit theorem for translation‐invariant operators on polygons

We prove Szeg?‐type trace asymptotics for translation‐invariant operators on polygons. More precisely, consider a Fourier multiplier on with a sufficiently decaying, smooth symbol . Let be the interior of a polygon and, for , define its scaled version . Then we study the spectral asymptotics for the operator , the spatial restriction of A onto : for entire functions h with we provide a complete asymptotic expansion of as . These trace asymptotics consist of three terms that reflect the geometry of the polygon. If P is replaced by a domain with smooth boundary, a complete asymptotic expansion of the trace has been known for more than 30 years. However, for polygons the formula for the constant order term in the asymptotics is new. In particular, we show that each corner of the polygon produces an extra contribution; as a consequence, the constant order term exhibits an anomaly similar to the heat trace asymptotics for the Dirichlet Laplacian.     

‐Theory for Schrödinger systems

In this article, we study for the ‐realization of the vector‐valued Schrödinger operator . Using a noncommutative version of the Dore–Venni theorem due to Monniaux and Prüss, we prove that the ‐realization of , defined on the intersection of the natural domains of the differential and multiplication operators which form , generates a strongly continuous contraction semigroup on . We also study additional properties of the semigroup such as extension to L¹, positivity, ultracontractivity and prove that the generator has compact resolvent.     

The concentration behavior of ground state solutions for a critical fractional Schrödinger–Poisson system

In this paper, we study the following critical fractional Schrödinger–Poisson system where is a small parameter, and , is the fractional critical exponent for 3‐dimension, has a positive global minimum, and are positive and have global maximums. We obtain the existence of a positive ground state solution by using variational methods, and we determine a concrete set related to the potentials and Q as the concentration position of these ground state solutions as . Moreover, we consider some properties of these ground state solutions, such as convergence and decay estimate.     

Singular Hamiltonian elliptic systems with critical exponential growth in dimension two

We will focus on the existence of nontrivial solutions to the following Hamiltonian elliptic system where are numbers belonging to the interval [0, 2), V is a continuous potential bounded below on by a positive constant and the functions f and g possess exponential growth range established by Trudinger–Moser inequalities in Lorentz–Sobolev spaces. The proof involves linking theorem and a finite‐dimensional approximation.     

Existence and regularity of positive solutions of a degenerate elliptic problem

Existence and regularity of positive solutions of a degenerate elliptic Dirichlet problem of the form in Ω, on , where Ω is a bounded smooth domain in , , are obtained via new embeddings of some weighted Sobolev spaces with singular weights and . It is seen that and admit many singular points in Ω. The main embedding results in this paper provide some generalizations of the well‐known Caffarelli–Kohn–Nirenberg inequality.     

Starshapedness of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings

We study solutions of the problem (0.1) where are open sets such that , , and f is a nonlinearity. Under different assumptions on f we prove that, if D₀ and D₁ are starshaped with respect to the same point , then the same occurs for every superlevel set of u.     

Scaling of spectra of a class of self‐similar measures on R

Let be two positive integers. For , let the self‐similar measure be defined by . It is known [18] that is a spectral measure with a spectrum where . In this paper, we give some conditions on under which the scaling set is also a spectrum of .     

Variable Hardy–Lorentz spaces

Let be a measurable function on with . We introduce the variable Hardy–Lorentz space for via the radial grand maximal function. Under the assumption that satisfies the log‐Hölder condition, we establish a version of Fefferman–Stein vector‐valued inequality in variable Lorentz space by interpolation. We also construct atomic decompositions for , and develop a theory of real interpolation and formulate the dual space of the variable Hardy–Lorentz space with and . As a byproduct, we obtain a new John–Nirenberg theorem. Furthermore, we get equivalent characterizations of the variable Hardy–Lorentz space by means of the Lusin area function, the Littlewood–Paley g‐function and the Littlewood–Paley ‐function. Finally, we investigate the boundedness of singular operators on for and .     

Recurrence of singularities for second order isotropic pseudodifferential operators

Let H be a self‐adjoint isotropic elliptic pseudodifferential operator of order 2. Denote by the solution of the Schrödinger equation with initial data . If u₀ is compactly supported the solution is smooth for small , but not for all t. We determine the wavefront set of in terms of the wavefront set of u₀ and the principal and subprincipal symbol of H.     

Spectral statistics of random Schrödinger operator with growing potential

In this work we investigate the spectral statistics of random Schrödinger operators acting on where are i.i.d random variables distributed uniformly on [0,1].     

Fractional Kirchhoff problem with critical indefinite nonlinearity

We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form where is a smooth bounded domain, and . Here M is the Kirchhoff coefficient and is the fractional critical Sobolev exponent. The parameter λ is positive and the is a real valued continuous function which is allowed to change sign. By using a variational approach based on the idea of Nehari manifold technique, we combine effects of a sublinear and a superlinear term to prove our main results.     

Isomorphisms of spaces with large distortion

Let K and S be locally compact Hausdorff spaces and let X be a strictly convex Banach space of finite dimension at least 2. In this paper, we prove that if there exists an isomorphism T from onto satisfying then K and S are homeomorphic. Here denotes the Schäffer constant of X. Even for the classical cases , and , this result is the X‐valued Banach–Stone theorem via isomorphism with the largest distortion that is known so far, namely . On the other hand, it is well known that this result is not true for , even though K and S are compact Hausdorff spaces.     

Lindelöf theorems for monotone Sobolev functions in Orlicz spaces on uniform domains

In this paper, we are concerned with Lindelöf type theorems for monotone (in the sense of Lebesgue) Sobolev functions u on a uniform domain satisfying where ? denotes the gradient, denotes the distance from z to the boundary , φ is of log‐type and ω is a weight function satisfying the doubling condition.     

Optimal decay for plates with rotational inertia and memory

We study the asymptotic behavior of a linear plate equation with effects of rotational inertia and a fractional damping in the memory term: where and the kernel g is exponentially decreasing. The main result of this work is the polynomial decay of their solutions when . We prove that the solutions decay with the rate and also that the decay rate is optimal. Furthermore, when , we obtain the exponential decay of the solutions.     

Smoothable zero dimensional schemes and special projections of algebraic varieties

We study the degrees of generators of the ideal of a projected Veronese variety to depending on the center of projection. This is related to the geometry of zero dimensional schemes of length 8 in , Cremona transforms of , and the geometry of Tonoli Calabi‐Yau threefolds of degree 17 in .