Multiple solutions for elliptic systems with nonlinearities of arbitrary growth

We prove the existence of infinitely many solutions for symmetric elliptic systems with nonlinearities of arbitrary growth. Moreover, if the symmetry of the problem is broken by a small enough perturbation term, we find at least three solutions. The proofs utilise a variational setting given by de Figueiredo and Ruf in order to prove an existence's result and the “algebraic” approach based on the Pohozaev's fibering method.     

Elliptic Equations and Systems with Subcritical and Critical Exponential Growth Without the Ambrosetti–Rabinowitz Condition

In this paper, we prove the existence of nontrivial nonnegative solutions to a class of elliptic equations and systems which do not satisfy the Ambrosetti–Rabinowitz (AR) condition where the nonlinear terms are superlinear at 0 and of subcritical or critical exponential growth at ∞. The known results without the AR condition in the literature only involve nonlinear terms of polynomial growth. We will use suitable versions of the Mountain Pass Theorem and Linking Theorem introduced by Cerami (Istit. Lombardo Accad. Sci. Lett. Rend. A, 112(2):332–336, 1978 Ann. Mat. Pura Appl., 124:161–179, 1980). The Moser–Trudinger inequality plays an important role in establishing our results. Our theorems extend the results of de Figueiredo, Miyagaki, and Ruf (Calc. Var. Partial Differ. Equ., 3(2):139–153, 1995) and of de Figueiredo, do Ó, and Ruf (Indiana Univ. Math. J., 53(4):1037–1054, 2004) to the case where the nonlinear term does not satisfy the AR condition. Examples of such nonlinear terms are given in Appendix A. Thus, we have established the existence of nontrivial nonnegative solutions for a wider class of nonlinear terms.     

On a semilinear elliptic problem without (PS) condition

An existence result for semilinear elliptic problems whose associated functionals do not satisfy a Palais-Smale condition is proved. The nonlinearity of our problem fits none of the conditions in Ambrosetti and Rabinowitz (J. Funct. Anal. 14 (1973) 349), de Figueiredo et al. (J. Math. Pures Appl. 61 (1982) 41) and Gidas and Spruck (Comm. Part. Diff. Eq. 6 (1981) 883). Some truncation happens to be essential, and in the argument some new results on Liouville-type theorems are established.     

Subordinate solutions and spectral measures of canonical systems

The theory of 2×2 trace-normed canonical systems of differential equations on ?⁺ can be put in the framework of abstract extension theory, cf. [9]. This includes the theory of strings as developed by I.S. Kac and M.G. Kre?n. In the present paper the spectral properties of such canonical systems are characterized by means of subordinate solutions. The theory of subordinacy for Schrödinger operators on the halfline ?⁺, was originally developed D.J. Gilbert and D.B. Pearson. Its extension to the framework of canonical systems makes it possible to describe the spectral measure of any Nevanlinna function in terms of subordinate solutions of the corresponding trace-normed canonical system, which is uniquely determined by a result of L. de Branges.     

Remarks on the Extremal Functions for the Moser-Trudinger Inequality

We will show in this paper that if A is very close to 1, then I（M,λ,m） =supu∈H0^1,n（m）,∫m｜△↓u｜^ndV=1∫Ω（e^αn｜u｜^n/（n-1）-λm∑k=1｜αnun/（n-1）｜k/k!）dV can be attained, where M is a compact-manifold with boundary. This result gives a counter-example to the conjecture of de Figueiredo and Ruf in their paper titled ＂On an inequality by Trudinger and Moser and related elliptic equations＂ （Comm. Pure. Appl. Math., 55, 135-152, 2002）.     

Infinitely many solutions for nonlocal elliptic systems of (p1(x),⋯,pn(x))‐Kirchhoff type

In this paper, we obtain the existence of infinitely solutions for a class of nonlocal elliptic systems of (p₁(x),?,p_n(x))‐Kirchhoff type. Our main results are new. Our approach are based on general variational principle because of B. Ricceri and the theory of the variable exponent Sobolev spaces. Copyright © 2015 John Wiley & Sons, Ltd.     

Partial regularity for nonlinear elliptic systems with continuous growth exponent

For weak solutions of nonlinear elliptic systems of the type ${- {\rm div}a(x, u(x), Du(x)) = 0,}$ with nonstandard p(x) growth, we show interior partial Hölder continuity for any Hölder exponent ${\alpha \in (0,1)}$ , provided that the exponent function is ‘logarithmic Hölder continuous’. The result also covers the up to now open partial regularity for systems with constant growth with exponent p less than two in the case of merely continuous dependence on the spacial variable x.     

On the Transmission of Riemann's Ideas to Portugal

Henrique Manual de Figueiredo is a secondary figure in the history of science in Portugal. His name is not recorded in any of the main biographical encyclopedias on his country, in international compilations such as the detailed work of May (Bibliography and Research Manual of the History of Mathematics, Toronto: Univ. of Toronto Press, 1973), or in studies on the history of science and culture in his country. However, he deserves to be remembered as a unique pioneer in the transmission to Portugal of Riemann's work, in particular of Riemann surfaces and the theory of algebraic curves. Although trained within the French tradition, and on friendly terms with French scientists till the end of his life, Figueiredo as a young man turned in the direction of the mathematical ideas then being developed in Germany. His life and work are also interesting from the point of view of the study of the transmission of science to and within peripheral countries and of their choice of foreign models. They suggest that, far from being a slow process of regular diffusion, the transmission of mathematical ideas from leading to peripheral mathematical communities is a complex process with selective sharp advances. Figueiredo was a respected mathematician within the structures of his own country, a professor at the University of Coimbra, who held several official positions in his country and represented it at one of the first international encounters involving science and technology in which peripheral countries took an active participation: the Paris Universal Exhibition of 1900Copyright 1999 Academic Press.Henrique Manuel de Figueiredo é uma figura de segundo plano na história da ciência em Portugal. Não se encontra qualquer referência ao seu nome, quer nas principais enciclopédias biográficas do seu paı́s, quer em publicações internacionais, tais como o minucioso trabalho de May (Bibliography and Research Manual of the History of Mathematics, Toronto: Univ. of Toronto Press, 1973), nem mesmo em estudos sobre a história da ciência e cultura do seu paı́s. Contudo, ele merece ser recordado como pioneiro na divulgação, em Portugal, do trabalho de Riemann, em particular superficies de Riemann e teoria das curvas algébraicas. Apesar da sua formação na escola francesca e de ter mantido laços de amizade com cientistas franceses, durante toda a sua vida, Figueiredo enquanto jovem deixou-se influenciar pelas ideias matemáticas então desenvolvidas na Alemanha. Na sua vida e obra tiveram também um papel importante a divulgação de ciência em paı́ses periféricos e o contributo para a escolha de modelos estrangeiros. Parece que a transmissão das ideias matemáticas dos centros principais para as comunidades matematicas periféricas, longe de ser um processo lento e regular, foi antes um processo complexo com progressos altamente irregulares. Figueiredo foi um matemático conceituado nas estruturas do seu próprio paı́s, era Professor na Universidade de Coimbra, ocupou vários cargos oficiais no seu paı́s e representou-o num dos primeiros encontros internacionais de ciência e tecnologia, a Exposição Universal de Paris, em 1900, na qual paı́ses periféricos tiveram uma participação activa.Copyright 1999 Academic Press.MSC 1991 subject classifications: 01A55; 01A60; 01A70.     

Gradient regularity for elliptic equations in the Heisenberg group

We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised in [J.J. Manfredi, G. Mingione, Regularity results for quasilinear elliptic equations in the Heisenberg group, Math. Ann. 339 (2007) 485-544], where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven in [A. Domokos, J.J. Manfredi, C1,α-regularity for p-harmonic functions in the Heisenberg group for p near 2, in: Contemp. Math., vol. 370, 2005, pp. 17-23]. In turn, using some recent techniques of Caffarelli and Peral [L. Caffarelli, I. Peral, On W1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998) 1-21], the a priori estimates found are shown to imply the suitable local Calderón-Zygmund theory for the related class of non-homogeneous, possibly degenerate equations involving discontinuous coefficients. These last results extend to the sub-elliptic setting a few classical non-linear Euclidean results [T. Iwaniec, Projections onto gradient fields and Lp-estimates for degenerated elliptic operators, Studia Math. 75 (1983) 293-312; E. DiBenedetto, J.J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993) 1107-1134], and to the non-linear case estimates of the same nature that were available in the sub-elliptic setting only for solutions to linear equations.     

The obstacle problem for nonlinear elliptic equations with variable growth and <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-data

The aim of this paper is twofold: to prove, for L ¹-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth, and to show some convergence and stability properties of the corresponding coincidence set. The latter follow from extending the Lewy-Stampacchia inequalities to the general framework of L ¹. Current address: Manel Sanchón, Universitat de Barcelona, Departament de Matemàtica Aplicada i Anàlisi, Gran Via 585, 08007 Barcelona, Spain; e-mail: msanchon@maia.ub.es Authors’ addresses: J. F. Rodrigues, CMUC, Department of Mathematics, University of Coimbra, and FCUL/Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal; M. Sanchón and J. M. Urbano, CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal     

On the local gevrey and quasi-analytic hypoellipticity of □b

This paper contains a variational treatment of the Ambrosetti–Prodi problem, including the superlinear case. The main result extends previous ones by Kazdan–Warner, Amann–Hess, Dancer, K. C. Chang and de Figueiredo. The required abstract results on critical point theory of functionals in Hilbert space are all proved using Ekeland's variational principle. These results apply as well to other superlinear elliptic problems provided an ordered pair of a sub– and a supersolution is exhibited.     

A note on the null condition for quadratic nonlinear Klein‐Gordon systems in two space dimensions

We consider the Cauchy problem for quadratic nonlinear Klein‐Gordon systems in two space dimensions with masses satisfying the resonance relation. Under the null condition in the sense of J.‐M. Delort, D. Fang, and R. Xue (J. Funct. Anal. 211 (2004), no. 2, 288–323), we show the global existence of asymptotically free solutions if the initial data are sufficiently small in some weighted Sobolev space. Our proof is based on an algebraic characterization of nonlinearities satisfying the null condition. © 2012 Wiley Periodicals, Inc.     

Conflict-directed A and its role in model-based embedded systems

Artificial Intelligence has traditionally used constraint satisfaction and logic to frame a wide range of problems, including planning, diagnosis, cognitive robotics and embedded systems control. However, many decision making problems are now being re-framed as optimization problems, involving a search over a discrete space for the best solution that satisfies a set of constraints. The best methods for finding optimal solutions, such as A^*, explore the space of solutions one state at a time. This paper introduces conflict-directed A^*, a method for solving optimal constraint satisfaction problems. Conflict-directed A^* searches the state space in best first order, but accelerates the search process by eliminating subspaces around each state that are inconsistent. This elimination process builds upon the concepts of conflict and kernel diagnosis used in model-based diagnosis [J. de Kleer, B.C. Williams, Diagnosing multiple faults, Artif. Intell. 32(1) (1987) 97-130; J. de Kleer, A. Mackworth, R. Reiter, Characterizing diagnoses and systems, Artif. Intell. 56 (1992) 197-222] and in dependency-directed search [R. Stallman, G.J. Sussman, Forward reasoning and dependency-directed backtracking in a system for computer-aided circuit analysis, Artif. Intell. 9 (1977) 135-196; J. Gaschnig, Performance measurement and analysis of certain search algorithms, Technical Report CMU-CS-79-124, Carnegie-Mellon University, Pittsburgh, PA, 1979; J. de Kleer, B.C. Williams, Back to backtracking: controlling the ATMS, in: Proceedings of AAAI-86, 1986, pp. 910-917; M. Ginsberg, Dynamic backtracking, J. Artif. Intell. Res. 1 (1993) 25-46]. Conflict-directed A^* is a fundamental tool for building model-based embedded systems, and has been used to solve a range of problems, including fault isolation [J. de Kleer, B.C. Williams, Diagnosing multiple faults, Artif. Intell. 32(1) (1987) 97-130], diagnosis [J. de Kleer, B.C. Williams, Diagnosis with behavioral modes, in: Proceedings of IJCAI-89, 1989, pp. 1324-1330], mode estimation and repair [B.C. Williams, P. Nayak, A model-based approach to reactive self-configuring systems, in: Proceedings of AAAI-96, 1996, pp. 971-978], model-compilation [B.C. Williams, P. Nayak, A reactive planner for a model-based executive, in: Proceedings of IJCAI-97, 1997] and model-based programming [M. Ingham, R. Ragno, B.C. Williams, A reactive model-based programming language for robotic space explorers, in: Proceedings of ISAIRAS-01, 2001].     

Higher integrability for nonlinear elliptic equations with variable growth and discontinuous coefficients

In this paper, based on the theory of variable exponent spaces, we study the higher integrability for a class of nonlinear elliptic equations with variable growth and discontinuous coefficients. Under suitable assumptions, we obtain a local gradient estimate in Orlicz space for weak solution.     

Local Algebras of Differential Operators

There is an increasing literature devoted to the study of boundary value problems using singularity theory. The resulting differential operators are typically Fredholm with index 0, defined on infinite-dimensional spaces, and they have often led to folds, cusps, and even higher-order Morin singularities. In this paper we develop some of the local algebras of germs of such differential Fredholm operators, extending the theory of the finite-dimensional case. We apply this work to nonlinear elliptic boundary value problems: in particular, we make further progress on a question proposed and initially studied by Ruf [1999, J. Differential Equations 151, 111-133]. We also make comments on several problems raised by others.     

Positive solutions of linear elliptic equations with critical growth in the Neumann boundary condition

We study the existence of positive solutions of a linear elliptic equation with critical Sobolev exponent in a nonlinear Neumann boundary condition. We prove a result which is similar to a classical result of Brezis and Nirenberg who considered a corresponding problem with nonlinearity in the equation. Our proof of the fact that the dimension three is critical uses a new Pohoaev-type identity.AMS Subject Classification: Primary: 35J65; Secondary: 35B33.     

Nonlinear Degenerate Oblique Boundary Value Problems for Second Order Fully Nonlinear Elliptic Equations

In this paper we study the existence theorem for solution of the nonlinear degenerate oblique boundary value problems for second order fully nonlinear elliptic equations F(x, u, Du, D²u) = 0 \quad x ∈ Ω, G(x, u, D, u) = 0, \qquad x ∈ ∂Ω where F (x, z, p, r) satisfies the natural structure conditions, G (x, z, q) satisfies G_q ≥ 0, G_x ≤ - G_0 < 0 and some structure conditions, vector τ is nowhere tangential to ∂Ω. This result extends the works of Lieberman G. M., Trudinger N. S. [2], Zhu Rujln [1] and Wang Feng [6].     

On elliptic and parabolic systems with x‐dependent multivalued graphs

We consider elliptic and parabolic systems with multivalued x‐dependent graphs. The existence of solutions for elliptic equation was established in (Ann. Inst. H. Poincare Anal. Non Linéaire 1990; 7 (3):123–160; Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 2004; 7 (1):23–59). We extend this result to the elliptic and parabolic systems, in particular to the systems describing a flow of non‐Newtonian incompressible fluids. Contrary to these two papers we follow the spirit of the compactness method of J. L. Lions for variational‐type operators, however, expanded on the framework of measure‐valued solutions. The main concept consists in applying the relation between x‐dependent maximal monotone graphs and 1‐Lipschitz Carathéodory functions to introduce the generalized Young measures. The method was announced in the short note (C. R. Math. Acad. Sci. Paris 2005; 340 (7):489–492). Copyright © 2006 John Wiley & Sons, Ltd.