Advertisement

Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

L p-estimates for the solutions of second order elliptic equations

  • 163 Accesses

  • 1 Citations

Summary

We investigate the homogeneous Dirichlet problem in H2,p for a second order elliptic partial differential equation in nondivergence form Lu=f in the case in which the leading coefficients of L belong to H1,n(Ω), Ω ⊂ Rn. We prove that if p belongs to a suitable neighbourhood of 2, then the above problem, has a unique solution u satisfying ∥D2u∥p⩽ C∥f∥p; furthermore, if f ε Hk,p, k=1,2, ..., and the coefficients of L satisfy some natural conditions, then the solution satisfies\(\left\| u \right\|_{H^{k + 2,p} } \leqslant C\left\| f \right\|_{H^{k,p} }\).

References

  1. [1]

    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.

  2. [2]

    S. Campanato,Un risultato relativo ad equazioni ellittiche del secondo ordine di tipo non variazionale, Ann. Sc. Norm. Sup. Pisa,21 (1967), pp. 701–707.

  3. [3]

    M. Chicco,Solvability of the Dirichlet problem in H 2,p(Ω) for a class of linear second order elliptic partial differential equations, Boll. UMI,4 (1971), pp. 374–387.

  4. [4]

    Fang-Hua Lin,Second derivative L p estimates for elliptic equations of nondivergent type, Proc. Am. Math. Soc.,96 (1986), pp. 447–451.

  5. [5]

    P. Manselli,A nonexistence and nonuniqueness example in Sobolev spaces for elliptic equations in nondivergence form, Boll. UMI,17-A (1980), pp. 302–306.

  6. [6]

    N. Meyers,An L p estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Sup. Pisa,17 (1963), pp. 189–206.

  7. [7]

    C. Miranda,Alcune osservazioni sulla maggiorazione in L v delle soluzioni deboli delle equazioni ellittiche del secondo ordine, Ann. Mat. Pura e Appl.,61 (1963), pp. 151–169.

  8. [8]

    C. Miranda,Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui, Ann. Mat. Pura e Appl.,63 (1963), pp. 353–386.

  9. [9]

    C.Pucci,Equazioni ellittiche con soluzioni in W2,p,p < 2, Convegno sulle equazioni alle derivate parziali, Bologna (1967), pp. 145–148.

  10. [10]

    C. Pucci,Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura e Appl.,74 (1966), pp. 15–30.

  11. [11]

    J. Serrin,Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Sup. Pisa,18 (1964), pp. 385–387.

  12. [12]

    G. Talenti,Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura e Appl.,69 (1965), pp. 285–304.

  13. [13]

    G. M. Troianiello,Elliptic differential equations and obstacle problems, The Univ. Series in Math., J. J. Kohn, Plenum Press, New York and London (1987).

  14. [14]

    M. Venturino,Sull'appartenenza ad H3(Ω)delle soluzioni di una classe di equazioni ellittiche a coefficienti discontinui, Analisi funzionale e applicazioni, Suppl. B.U.M.I., Vol. I (1980), 197–218.

Download references

Author information

Additional information

Lavoro eseguito nell'ambito del gruppi 40% e 60% del M.P.I.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mandras, F., Porru, G. L p-estimates for the solutions of second order elliptic equations. Annali di Matematica pura ed applicata 156, 253–263 (1990). https://doi.org/10.1007/BF01766983

Download citation

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Natural Condition
  • Unique Solution
  • Elliptic Equation