Infinitely Many Solutions for a Class of Fractional Boundary Value Problems with Dirichlet Boundary Conditions

In this paper we prove the existence and multiplicity of weak solutions for a class of fractional boundary value problem. Our approach is based on a critical point result contained in Bonanno and Molica Bisci [Bound. Value. Probl. 2009, 1–20 (2009)].     

Applications of Fourier Analysis in Homogenization of Dirichlet Problem III: Polygonal Domains

In this paper we prove convergence results for the homogenization of the Dirichlet problem for elliptic equations in divergence form with rapidly oscillating boundary data and non oscillating coefficients in convex polygonal domains. Our analysis is based on integral representation of solutions. Under a certain Diophantine condition on the boundary of the domain and smooth coefficients we prove pointwise, as well as \(L^p\) convergence results. For larger exponents \(p\) we prove that the \(L^p\) convergence rate is close to optimal. We also suggest several directions of possible generalization of the results in this paper.     

Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit

We prove that any weak space-time \(L^2\) vanishing viscosity limit of a sequence of strong solutions of Navier–Stokes equations in a bounded domain of \({\mathbb R}^2\) satisfies the Euler equation if the solutions’ local enstrophies are uniformly bounded. We also prove that \(t-a.e.\) weak \(L^2\) inviscid limits of solutions of 3D Navier–Stokes equations in bounded domains are weak solutions of the Euler equation if they locally satisfy a scaling property of their second-order structure function. The conditions imposed are far away from boundaries, and wild solutions of Euler equations are not a priori excluded in the limit.     

Necessary condition for eigensolutions of the 1-Laplace operator by means of inner variations

Eigenfunctions of the $p$ -Laplace operator for $p>1$ are defined to be critical points of an associated variational problem or, equivalently, to be solutions of the corresponding Euler–Lagrange equation. In the highly degenerated limit case of the 1-Laplace operator eigenfunctions can also be defined to be critical points of the corresponding variational problem if critical points are understood on the basis of the weak slope. However, the associated Euler–Lagrange equation has many solutions that are not critical points and, thus, it cannot be used for an equivalent definition. The present paper provides a new necessary condition for eigenfunctions of the 1-Laplace operator by means of inner variations of the associated variational problem and it is shown that this condition rules out certain solutions of the Euler–Lagrange equation that are not eigenfunctions.     

A Multiplicity Result for an Elliptic Anisotropic Differential Inclusion Involving Variable Exponents

In this paper we are concerned with the study of a class of quasilinear elliptic differential inclusions involving the anisotropic $\overrightarrow {p}(\cdot)$ -Laplace operator, on a bounded open subset of ${\mathbb R}^n$ which has a smooth boundary. The abstract framework required to study this kind of differential inclusions lies at the interface of three important branches in analysis: nonsmooth analysis, the variable exponent Lebesgue–Sobolev spaces theory and the anisotropic Sobolev spaces theory. Using the concept of nonsmooth critical point we are able to prove that our problem admits at least two non-trivial weak solutions.     

On the pure critical exponent problem for the p-Laplacian

In this paper we prove existence and multiplicity of positive and sign-changing solutions to the pure critical exponent problem for the $p$ -Laplacian operator with Dirichlet boundary conditions on a bounded domain having nontrivial topology and discrete symmetry. Pioneering works related to the case $p=2$ are Brezis and Nirenberg (Comm Pure Appl Math 36, 437–477, 1983), Coron (C R Acad Sci Paris Sr I Math 299, 209–212, 1984), and Bahri and Coron (Comm. Pure Appl. Math. 41, 253–294, 1988). A global compactness analysis is given for the Palais-Smale sequences in the presence of symmetries.     

Very weak solutions for the Stokes problem in an exterior domain

In this paper, we study the Stokes problem in exterior domain of ${\mathbb{R}^{3}}$ . We are interested in the existence and the uniqueness of very weak solutions. Here, we extend a result proved by Farwig et al. (J Math Soc Japan 59(1):127–150, 2007) and we prove the existence and the uniqueness of a second type of very weak solution.     

Nonstationary Stokes system in Sobolev–Slobodetski spaces

We consider an initial-boundary value problem for the nonstationary Stokes system in a bounded domain $\Omega \subset \mathbb R ^3$ with slip boundary conditions. We prove the existence in the Hilbert–Sobolev–Slobodetski spaces with fractional derivatives. The proof is divided into two main steps. In the first step by applying the compatibility conditions an extension of initial data transforms the considered problem to a problem with vanishing initial data such that the right-hand sides data functions can be extended by zero on the negative half-axis of time in the above mentioned spaces. The problem with vanishing initial data is transformed to a functional equation by applying an appropriate partition of unity. The existence of solutions of the equation is proved by a fixed point theorem. We prove the existence of such solutions that $v\in H^{l+2,l/2+1}(\Omega \times (0,T)),\,\nabla p\in H^{l,l/2}(\Omega \times (0,T)),\,v$ —velocity, $p$ —pressure, $l\in \mathbb R _+\cup \{0\},\,l \ne [l]+\frac{1}{2}$ and the spaces are introduced by Slobodetski and used extensively by Lions–Magenes. We should underline that to show solvability of the Stokes system we need only solvability of the heat and the Poisson equations in $\mathbb R ^3$ and $\mathbb R _+^3$ . This is possible because the slip boundary conditions are considered.     

Regularity of the Solution to Nonlinear Anisotropic Elliptic Equations with Variable Exponents and Irregular Data

In this paper, we prove the existence and regularity of weak solutions for a class of nonlinear anisotropic elliptic equations with \(p_i(x)\) growth conditions and \(L^m\) data, with m being small. The functional setting involves Lebesgue–Sobolev spaces with variable exponents. Our results are generalizations of the corresponding results in the constant exponent case and some results given in Bendahmane et al. (Commun Pure Appl Anal 12:1201–1220, 2013).     

REGULARITY OF THE FREE BOUNDARY FOR SOME ELLIPTIC AND PARABOLIC PROBLEMS. II

In this paper we extend the results of the first one to solutions of some obstacle problem in the semilinear elliptic case that are used as a model for a gas problem. More precisely, we prove that the points of the free boundary, where the zero set has no density, lie in a Lipschitz surface. Furthermore, we get the C ¹ regularity for singular points with some (n ? 1)-density.

We also investigate the free boundary at points with density. We show that the set of these points is locally a C ¹ surface. This result is an extension of those achieved by Alt and Phillips [3] Alt, H. W. and Phillips, D. 1986. A free boundary problem for semilinearelliptic equations. J. Reine Angew. Math., 368: 63–107.  [Google Scholar], where it is used a concept stronger than the “density” applied here.     

Inviscid Incompressible Limits Under Mild Stratification: A Rigorous Derivation of the Euler–Boussinesq System

We consider the full Navier–Stokes–Fourier system in the singular regime of small Mach and large Reynolds and Péclet numbers, with ill prepared initial data on an unbounded domain \(\Omega \subset R^3\) with a compact boundary. We perform the singular limit in the framework of weak solutions and identify the Euler–Boussinesq system as the target problem.     

The oblique derivative problem for nonlinear elliptic complex equations of second order in multiply connected unbounded domains

In this article,we discuss that an oblique derivative boundary value problem for nonlinear uniformly elliptic complex equation of second order with the boundary conditions in a multiply connected unbounded domain D.The above boundary value problem will be called Problem P.Under certain conditions,by using the priori estimates of solutions and Leray-Schauder fixed point theorem,we can obtain some results of the solvability for the above boundary value problem(0.1) and(0.2).     

A Mountain Pass-type Theorem for Vector-valued Functions

The mountain pass theorem for scalar functionals is a fundamental result of the minimax methods in variational analysis. In this work we extend this theorem to the class of \(\mathcal{C}^{1}\) functions \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}\), where the image space is ordered by the nonnegative orthant \(\mathbb{R}_{+}^{m}\). Under suitable geometrical assumptions, we prove the existence of a critical point of f and we localize this point as a solution of a minimax problem. We remark that the considered minimax problem consists of an inner vector maximization problem and of an outer set-valued minimization problem. To deal with the outer set-valued problem we use an ordering relation among subsets of \(\mathbb{R}^{m}\) introduced by Kuroiwa. In order to prove our result, we develop an Ekeland-type principle for set-valued maps and we extensively use the notion of vector pseudogradient.     

Planar least gradient problem: existence,regularity and anisotropic case

We show existence of solutions to the least gradient problem on the plane for boundary data in \(BV(\partial \varOmega )\). We also provide an example of a function \(f \in L^1(\partial \varOmega ) \backslash \) \((C(\partial \varOmega ) \cup BV(\partial \varOmega ))\), for which the solution exists. We also show non-uniqueness of solutions even for smooth boundary data in the anisotropic case for a nonsmooth anisotropy. We additionally prove a regularity result valid also in higher dimensions.     

Hypoellipticity for infinitely degenerate quasilinear equations and the dirichlet problem

In [10], we considered a class of infinitely degenerate quasilinear equations of the form div $A(x,w)\nabla w + \overrightarrow r (x,w) \cdot \nabla w + f(x,w) = 0$ and derived a priori bounds for high order derivatives D ^a w of their solutions in terms of w and ?w. We now show that it is possible to obtain bounds in terms of just w for a further subclass of such equations, and we apply the resulting estimates to prove that continuous weak solutions are necessarily smooth. We also obtain existence, uniqueness, and interior ${\varrho ^\infty }$ -regularity of solutions for the corresponding Dirichlet problem with continuous boundary data.     

Regularity for elliptic systems of differential forms and applications

We prove existence and up to the boundary regularity estimates in \(L^{p}\) and Hölder spaces for weak solutions of the linear system

$$\begin{aligned} \delta \left( A d\omega \right) + B^{T}d\delta \left( B\omega \right) = \lambda B\omega + f \text { in } \varOmega , \end{aligned}$$

with either \( \nu \wedge \omega \) and \(\nu \wedge \delta \left( B\omega \right) \) or \(\nu \lrcorner B\omega \) and \(\nu \lrcorner \left( A d\omega \right) \) prescribed on \(\partial \varOmega .\) The proofs are in the spirit of ‘Campanato method’ and thus avoid potential theory and do not require a verification of Agmon–Douglis–Nirenberg or Lopatinski?–Shapiro type conditions. Applications to a number of related problems, such as general versions of the time-harmonic Maxwell system, stationary Stokes problem and the ‘div-curl’ systems, are included.

     

An existence result for a class of quasilinear elliptic eigenvalue problems in unbounded domains

We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p–Laplacian operator and subcritical nonlinearities satisfying Ambrosetti–Rabinowitz type conditions. Using Morse theory and a cohomological local splitting as in Degiovanni et al. (Commun Contemp Math 12:475–486, 2010), we prove the existence of a nontrivial weak solution for all (real) values of the eigenvalue parameter. Our result is new even in the semilinear case p = 2 and complements some recent results obtained in Autuori et al. (Adv Anal Equ 18:1–48, 2013).     

On the Riemann Boundary Value Problem for Null Solutions to Iterated Generalized Cauchy–Riemann Operator in Clifford Analysis

In this paper we consider a kind of Riemann boundary value problem (for short RBVP) for null solutions to the iterated generalized Cauchy–Riemann operator and the polynomially generalized Cauchy–Riemann operator, on the sphere of ${\mathbb{R}^{n+1}}$ with Hölder-continuous boundary data. Making full use of the poly-Cauchy type integral operator in Clifford analysis, we give explicit integral expressions of solutions to this kind of boundary value problems over the sphere of ${\mathbb{R}^{n+1}}$ . As special cases solutions of the corresponding boundary value problems for the classical poly-analytic and meta-analytic functions are also derived, respectively.     

Weak solutions of the stationary Navier–Stokes equations for a viscous incompressible fluid past an obstacle

Consider the stationary Navier–Stokes equations in an exterior domain $\varOmega \subset \mathbb{R }^3 $ with smooth boundary. For every prescribed constant vector $u_{\infty } \ne 0$ and every external force $f \in \dot{H}_2^{-1} (\varOmega )$ , Leray (J. Math. Pures. Appl., 9:1–82, 1933) constructed a weak solution $u $ with $\nabla u \in L_2 (\varOmega )$ and $u - u_{\infty } \in L_6(\varOmega )$ . Here $\dot{H}^{-1}_2 (\varOmega )$ denotes the dual space of the homogeneous Sobolev space $\dot{H}^1_{2}(\varOmega ) $ . We prove that the weak solution $u$ fulfills the additional regularity property $u- u_{\infty } \in L_4(\varOmega )$ and $u_\infty \cdot \nabla u \in \dot{H}_2^{-1} (\varOmega )$ without any restriction on $f$ except for $f \in \dot{H}_2^{-1} (\varOmega )$ . As a consequence, it turns out that every weak solution necessarily satisfies the generalized energy equality. Moreover, we obtain a sharp a priori estimate and uniqueness result for weak solutions assuming only that $\Vert f\Vert _{\dot{H}^{-1}_2(\varOmega )}$ and $|u_{\infty }|$ are suitably small. Our results give final affirmative answers to open questions left by Leray (J. Math. Pures. Appl., 9:1–82, 1933) about energy equality and uniqueness of weak solutions. Finally we investigate the convergence of weak solutions as $u_{\infty } \rightarrow 0$ in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case $u_{\infty } \ne 0$ .