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\(W^{2,n}\) a priori estimates for solutions to fully nonlinear equations. (English) Zbl 0792.35020

Let \(O(\lambda,\Lambda)\) be the class of fully nonlinear operators \(F(D^ 2u,x)\) satisfying, for some positive numbers \(\lambda\), \(\Lambda\) the usual ellipticity condition. For \(F \in O(\lambda,\Lambda)\) the following result is proved. Assume that solutions \(h\) to the Dirichlet problem \(F(D^ 2 h(x), y) = 0\) on the ball \(B_ r(y)\) and \(h = w\) on \(\partial B_ r(y)\) satisfy the estimate \[ \| h\|_{C^{1,1}(B_{r/2}(y))} \leq Xr^{-2}\| h\|_{L^ \infty(B_ r(y))} \] for all \(y \in B_ 1\). Then there exists \(\varepsilon > 0\) depending on \(\Lambda/\lambda\) and \(n\) such that if \(u\) is a solution to \(F(D^ 2 u,x) = f(x)\) on \(B_ 1\), where \(f\in L^ p(B_ 1)\), \(p > n-\varepsilon\) and \(F(D^ 2u, x)\) is “suitably close” to \(F(D^ 2h,y)\), the following estimation holds \[ \| u\|_{W^{2,p}(B_{1/2})} \leq C[\| u\|_{L^ \infty(B_ 1)} + \| f\|_{L^ p(B_ 1)}]. \] Such a result was proved by L. Caffarelli [Ann. Math., II. Ser. 130, No. 1, 189-213 (1989; Zbl 0692.35017)] in the same situation for \(p > n\) by using a similar method.
As an application, the author proves the following theorem: Let \(L_ m \in O(\lambda,\Lambda)\), \(L_ m u = a^ m_{ij} (x) D_{ij} u\), \(m = 1,2,\dots\). Assume that the modulus of continuity of the coefficients is independent of \(m\), \(f \in L^ p(B_ 1)\), \(p > n - \varepsilon\) and \(\phi \in C(\partial B_ 1)\). Consider the operator \(F(D^ 2u,x) = \sup_ m L_ m u(x)\). Then, there exists a unique \(u \in W^{2,p}_{\text{loc}}(B_ 1) \cap C(B_ 1)\) satisfying \(F(D^ 2 u,x) = f(x)\) a.e. and \(u = \phi\) on \(\partial B_ 1\).
Reviewer: G.Porru (Cagliari)

MSC:

35B45 A priori estimates in context of PDEs
35J60 Nonlinear elliptic equations

Citations:

Zbl 0692.35017
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