Summary
In this paper we study the problem:
where f is given inC ∞# ={g∈C∞\g peiodic}. We find, for every f∈C ∞# , at least one solution u in the same classC ∞# . The hypothesis on F(ξ, η) is the following:
moreover, when m=2, a Cordes condition is required:
The method used is based on R. Caccioppoli's inversion theorem for proper-open mappings, in the context of countably normed spaces. An essential tool is a theorem of S. Campanato concerning the existence of W2,p-solutions (p «near 2 ») for elliptic-Cordes equations with bounded measurable coefficients.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Bibliografia
R.Adams,Sobolev Spaces, Academic Press, 1975.
S.Agmon,Lectures on elliptic boundary value problems, D. Van Nostrand Company, Inc., 1965.
S. Agmon,The L p approach to the Dirichlet problem, Ann. Sc. N. Sup. Pisa,13 (1959), pp. 405–448.
S. Agmon -A. Douglis -L. Nirenberg,Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions (I), Comm. Pure Appl. Math.,12 (1959), pp. 623–727.
S. Bernstein,Sur la généralisation du problème de Dirichlet, Math. Ann.,62 (1906), pp. 253–271;69 (1910), pp. 82–136.
L.Bers - L.Nirenberg,On linear and nonlinear elliptic boundary value problems in the plane, Convegno Int. Eq. Lin. Der. Parz., Trieste (1954), pp. 141–167.
R. Caccioppoli,Un principio di inversione per le corrispondenze funzionali e sue appli- cazioni alle equazioni a derivate parziali, Rend. Acc. Lincei,16 (1932), pp. 390–395, 484–489.
S. Campanato,Un risultato relativo ad equazioni ellittiche del secondo ordine di tipo non variazionale, Ann. Sc. N. Sup. Pisa,21 (1967), pp. 701–707.
A. Douglis -L. Nirenberg,Interior Estimates for Elliptic Systems of Partial Differential Equations, Comm. Pure Appl. Math.,8 (1955), pp. 503–538.
A.Friedman,Generalized Functions and Partial Differential Equations, Prentice-Hall, Inc., 1963.
A. Friedman,On the Regularity of the Solution of Non-Linear Elliptic and Parabolic Systems of Partial Differential Equations, J. Math. Mech.,7 (1958), pp. 43–59.
A.Friedman,Partial Differential Equations, Holt, Rinehart and Winston, Inc., 1969.
E.Giusti,Equazioni ellittiche del secondo ordine, Quaderni dell'UMI, Pitagora Editrice, 1978.
I. M.Guelfand - G. E.Chilov,Les Distributions (tome 2), Dunod, 1964.
J. Leray,Majoration des dérivées secondes des solutions d'un problème de Dirichlet, Journ. de Math.,17, Fasc. I (1938), pp. 89–104.
J. Leray,Discussion d'un problème de Dirichlet, Journ. de Math.,18, Fasc. III (1939), pp. 249–284.
C.Miranda,Partial Differential Equations of Elliptic Type, Springer Verlag, 1970.
C.Miranda,Problemi di esistenza in analisi funzionale, Scuola Normale Superiore di Pisa, 1975.
C.Miranda,Istituzioni di analisi funzionale lineare, Unione Matematica Italiana, 1978.
G.Prodi - A.Ambrosetti,Analisi non lineare, Scuola Normale Superiore di Pisa, 1973.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Canfora, A. Sull'esistenza di soluzioni periodiche per certe equazioni ellittiche totalmente non lineari. Annali di Matematica pura ed applicata 128, 253–286 (1981). https://doi.org/10.1007/BF01789477
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01789477