$L^p$ estimates for degenerate elliptic systems with VMO coefficients
نویسندگان
چکیده
منابع مشابه
Parabolic and Elliptic Systems with Vmo Coefficients
We consider second order parabolic and elliptic systems with leading coefficients having the property of vanishing mean oscillation (VMO) in the spatial variables. An Lq −Lp theory is established for systems both in divergence and non-divergence form. Higher order parabolic and elliptic systems are also discussed briefly.
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An Lp-theory of divergence and non-divergence form elliptic and parabolic equations is presented. The main coefficients are supposed to belong to the class V MOx, which, in particular, contains all functions independent of x. Weak uniqueness of the martingale problem associated with such equations is obtained.
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− Dα(A i j (x, u)Dβu j) = Bi(x, u,Du), a. e. x ∈ Ω, i = 1, 2, . . . ,N; (1.1) where A(x, u) = (A i j (x, u)) is a VMO function in x ∈ Ω uniformly with respect to u ∈ RN and continuous in u uniformly with respect to x ∈ Ω, and Bi(x, u,Du) satisfies the controllable growth. In the context, we adopt Einstein’s convention by summing over repeated indices with α, β = 1, 2, . . . , n and i, j = 1, 2,...
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(1.1) −tr ( A(x)Du ) + λu+ |Du| = f(x) . Here Ω is an open bounded subset of R , N ≥ 2, and A : Ω 7→ S N is a bounded and continuous map into the space of symmetric non-negative matrices of order N , λ ≥ 0 and p > 1 are given numbers and f : Ω 7→ R is a continuous function. It is well-known that equations such as (1.1) arise in stochastic control as the HamiltonJacobi-Bellman equation satisfied...
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ژورنال
عنوان ژورنال: St. Petersburg Mathematical Journal
سال: 2014
ISSN: 1061-0022,1547-7371
DOI: 10.1090/s1061-0022-2014-01322-2