TY - JOUR

T1 - On the existence of W2 p solutions for fully nonlinear elliptic equations under either relaxed or no convexity assumptions

AU - Krylov, N. V.

PY - 2017/12/1

Y1 - 2017/12/1

N2 - We establish the existence of solutions of fully nonlinear elliptic second-order equations like H(v,Dv,D2v,x) = 0 in smooth domains without requiring H to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of H at points at which |D2v|≤ K, where K is any given constant. For large |D2v| some kind of relaxed convexity assumption with respect to D2v mixed with a VMO condition with respect to x are still imposed. The solutions are sought in Sobolev classes. We also establish the solvability without almost any conditions on H, apart from ellipticity, but of a "cut-off" version of the equation H(v,Dv,D2v,x) = 0.

AB - We establish the existence of solutions of fully nonlinear elliptic second-order equations like H(v,Dv,D2v,x) = 0 in smooth domains without requiring H to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of H at points at which |D2v|≤ K, where K is any given constant. For large |D2v| some kind of relaxed convexity assumption with respect to D2v mixed with a VMO condition with respect to x are still imposed. The solutions are sought in Sobolev classes. We also establish the solvability without almost any conditions on H, apart from ellipticity, but of a "cut-off" version of the equation H(v,Dv,D2v,x) = 0.

KW - Fully nonlinear elliptic equations

KW - cut-off equations

KW - finite differences

UR - http://www.scopus.com/inward/record.url?scp=85009999971&partnerID=8YFLogxK

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U2 - 10.1142/S0219199717500092

DO - 10.1142/S0219199717500092

M3 - Article

AN - SCOPUS:85009999971

VL - 19

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

IS - 6

M1 - 1750009

ER -