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Register a new userGreens matrices of second order elliptic systems with measurable coefficients in two dimensional domains
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We study Greens matrices for divergence form, second order strongly elliptic systems with bounded measurable coefficients in two dimensional domains. We establish existence, uniqueness, and pointwise estimates of the Greens matrices.
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We establish existence and various estimates of fundamental matrices and Greens matrices for divergence form, second order strongly parabolic systems in arbitrary cylindrical domains under the assumption that solutions of the systems satisfy an interior H{o}lder continuity estimate. We present a unified approach valid for both the scalar and the vectorial cases.
We consider a second-order parabolic equation in $bR^{d+1}$ with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Holder continuous in the space variables. We show that global Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the $L_{infty}$-norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case of bounded coefficients.
The solvability in $W^{2}_{p}(bR^{d})$ spaces is proved for second-order elliptic equations with coefficients which are measurable in one direction and VMO in the orthogonal directions in each small ball with the direction depending on the ball. This generalizes to a very large extent the case of equations with continuous or VMO coefficients.
We construct Greens functions for second order parabolic operators of the form $Pu=partial_t u-{rm div}({bf A} abla u+ boldsymbol{b}u)+ boldsymbol{c} cdot abla u+du$ in $(-infty, infty) times Omega$, where $Omega$ is an open connected set in $mathbb{R}^n$. It is not necessary that $Omega$ to be bounded and $Omega = mathbb{R}^n$ is not excluded. We assume that the leading coefficients $bf A$ are bounded and measurable and the lower order coefficients $boldsymbol{b}$, $boldsymbol{c}$, and $d$ belong to critical mixed norm Lebesgue spaces and satisfy the conditions $d-{rm div} boldsymbol{b} ge 0$ and ${rm div}(boldsymbol{b}-boldsymbol{c}) ge 0$. We show that the Greens function has the Gaussian bound in the entire $(-infty, infty) times Omega$.
We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which is assumed to be only measurable either in the time or the spatial variable. As functions of the other variables the coefficients have small bounded mean oscillation (BMO) semi-norms. The lower-order coefficients are allowed to blow up near the boundary with a certain optimal growth condition. As a corollary, we also obtain the corresponding results for elliptic equations.
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