Concentration-compactness principle for variable exponent spaces and applications.
Bonder, Julián Fernández, Silva, Analía (2010)
Electronic Journal of Differential Equations (EJDE) [electronic only]
Agarwal, Ravi P., Filippakis, Michael E., O'Regan, Donal, Papageorgiou, Nikolaos S. (2009)
Boundary Value Problems [electronic only]
Pierre Bousquet (2013)
ESAIM: Control, Optimisation and Calculus of Variations
We consider a nonlinear elliptic equation of the form div [a(∇u)] + F[u] = 0 on a domain Ω, subject to a Dirichlet boundary condition tru = φ. We do not assume that the higher order term a satisfies growth conditions from above. We prove the existence of continuous solutions either when Ω is convex and φ satisfies a one-sided bounded slope condition, or when ais radial: a ( ξ ) = l ( | ξ | ) | ξ | ξ for some increasingl:ℝ+ → ℝ+.
Francis Clarke (2005)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
We study the problem of minimizing over the functions that assume given boundary values on . The lagrangian and the domain are assumed convex. A new type of hypothesis on the boundary function is introduced: thelower (or upper) bounded slope condition. This condition, which is less restrictive than the familiar bounded slope condition of Hartman, Nirenberg and Stampacchia, allows us to extend the classical Hilbert-Haar regularity theory to the case of semiconvex (or semiconcave) boundary...
Ken Shirakawa (2009)
Banach Center Publications
In this paper, a one-dimensional Euler-Lagrange equation associated with the total variation energy, and Euler-Lagrange equations generated by approximating total variations with linear growth, are considered. Each of the problems presented can be regarded as a governing equation for steady-states in solid-liquid phase transitions. On the basis of precise structural analysis for the solutions, the continuous dependence between the solution classes of approximating problems and that of the limiting...
Aleksandra Orpel (2005)
Colloquium Mathematicae
We discuss the existence of solutions for a certain generalization of the membrane equation and their continuous dependence on function parameters. We apply variational methods and consider the PDE as the Euler-Lagrange equation for a certain integral functional, which is not necessarily convex and coercive. As a consequence of the duality theory we obtain variational principles for our problem and some numerical results concerning approximation of solutions.
Futoshi Takahashi (2014)
Mathematica Bohemica
We study the semilinear problem with the boundary reaction where , , is a smooth bounded domain, is a smooth, strictly positive, convex, increasing function which is superlinear at , and is a parameter. It is known that there exists an extremal parameter such that a classical minimal solution exists for , and there is no solution for . Moreover, there is a unique weak solution corresponding to the parameter . In this paper, we continue to study the spectral properties of and show...
Frank Pacard (1994)
Annales de l'I.H.P. Analyse non linéaire
Jaroslav Haslinger, Ivan Hlaváček (1976)
Aplikace matematiky
Ivan Hlaváček (1979)
Aplikace matematiky
An equilibrium triangular block-element, proposed by Watwood and Hartz, is subjected to an analysis and its approximability property is proved. If the solution is regular enough, a quasi-optimal error estimate follows for the dual approximation to the mixed boundary value problem of elasticity (based on Castigliano's principle). The convergence is proved even in a general case, when the solution is not regular.
Stefan Müller, Vladimír Šverák (1999)
Journal of the European Mathematical Society
We study solutions of first order partial differential relations , where is a Lipschitz map and is a bounded set in matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of and second we replace Gromov’s −convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our work was originally...
Samira Benmouloud-Sbai, Mohamed Guedda (2003)
Annales de la Faculté des sciences de Toulouse : Mathématiques
Marino Badiale, Enrico Serra (2004)
Revista Matemática Iberoamericana
Motivated by a problem arising in astrophysics we study a nonlinear elliptic equation in RN with cylindrical symmetry and with singularities on a whole subspace of RN. We study the problem in a variational framework and, as the nonlinearity also displays a critical behavior, we use some suitable version of the Concentration-Compactness Principle. We obtain several results on existence and nonexistence of solutions.
Michael Struwe (1988)
Annales de l'I.H.P. Analyse non linéaire
Andrea Malchiodi, Luca Martinazzi (2014)
Journal of the European Mathematical Society
On the unit disk we study the Moser-Trudinger functional and its restrictions , where for . We prove that if a sequence of positive critical points of (for some ) blows up as , then , and weakly in and strongly in . Using this fact we also prove that when is large enough, then has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.
Jaroslav Haslinger, Ivan Hlaváček (1975)
Aplikace matematiky
Chabrowski, J. (1996)
Portugaliae Mathematica
Bruno Canuto, Otared Kavian (2004)
Bollettino dell'Unione Matematica Italiana
For a bounded and sufficiently smooth domain in , , let and be respectively the eigenvalues and the corresponding eigenfunctions of the problem (with Neumann boundary conditions) We prove that knowledge of the Dirichlet boundary spectral data , determines uniquely the Neumann-to-Dirichlet (or the Steklov- Poincaré) map for a related elliptic problem. Under suitable hypothesis on the coefficients their identifiability is then proved. We prove also analogous results for Dirichlet...
Mugnolo, Delio, Romanelli, Silvia (2006)
Electronic Journal of Differential Equations (EJDE) [electronic only]
Lucio Boccardo (2008)
ESAIM: Control, Optimisation and Calculus of Variations
We present a revisited form of a result proved in [Boccardo, Murat and Puel, Portugaliae Math.41 (1982) 507–534] and then we adapt the new proof in order to show the existence for solutions of quasilinear elliptic problems also if the lower order term has quadratic dependence on the gradient and singular dependence on the solution.