Semigroups generated by elliptic operators in non-divergence form on C-o(Ω)

The [formula omitted] Dirichlet problem for second-order, non-divergence form operators: solvability and perturbation results

On every right circular cone and for every uniformly elliptic operator in nondivergence form there exists a barrier, that is, a supersolution which is zero at the vertex and positive elsewhere on the closed cone. These barriers are applied to the Dirichlet problem. Other applications are deferred to a later paper on « extremal barriers »

We give a detailed analysis of supersolutions of form rλf(θ) for the classLα of uniformly elliptic operators in nondivergence form with ellipticity constant α. Fundamental to the analysis is the extremal supersolution rλFλ, α(θ) for each real λ, which is itself a solution of a certain extremal elliptic equation and which remains positive on a cone of wider aperature than any other supersolution of this form. These supersolutions are employed as “barriers” to yeild Phragmen-Lindelof theorems (λ>0) for unbounded domains contained in a cone, and Holder continuity (λ>0) and removeable singularity (λ<0) results at boundary points with an exterior cone property. In each case the growth condition 0(rλ) involved, depending on the aperature of the cone, is best possible. Similar results carry through for operators with singular lower order terms and possibily positive zero order coefficient.

We prove Cordes type estimates for subelliptic linear partial differential operators in non-divergence form with measurable coefficients in the Heisenberg group. As an application we establish interior horizontal W 2 , 2 W^{2,2} -regularity for p-harmonic functions in the Heisenberg group H 1 {\mathbb H}^1 for the range 17 − 1 2 ≤ p &gt; 5 + 5 2 \frac {\sqrt {17}-1}{2} \leq p &gt; \frac {5+\sqrt {5}}{2} .

We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabr\'e proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabr\'e's result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature.

We consider non-negative solutions to a class of second-order degenerate Kolmogorov operators of the form L=Sigma(m)(i,j=1)a(i,j)(z)partial derivative(xixj)+Sigma(m)(i=1)a(i)(z)partial derivative(x ...

Potential analysis for a class of diffusion equations: A Gaussian bounds approach

Given an elliptic operator L=-div(A∇·) subject to mixed boundary conditions on an open subset of Rd, we study the relation between Gaussian pointwise estimates for the kernel of the associated heat semigroup, Hölder continuity of L-harmonic functions and the growth of the Dirichlet energy. To this end, we generalize an equivalence theorem of Auscher and Tchamitchian to the case of mixed boundary conditions and to open sets far beyond Lipschitz domains. Yet, we prove the consistency of our abstract result by encompassing operators with real-valued coefficients and their small complex perturbations into one of the aforementioned equivalent properties. The resulting kernel bounds open the door for developing a harmonic analysis for the associated semigroups on rough open sets.

Strict inequality of Robin eigenvalues for elliptic differential operators on Lipschitz domains

We construct the fundamental solutions for the sub-elliptic operators in non-divergence form∑i,jai,j(x,t)XiXj−∂t{\textstyle \sum _{i,j}} a_{i,j}(x,t)\,X_iX_j-\partial _tand∑i,jai,j(x)XiXj{\textstyle \sum _{i,j}}a_{i,j}(x)\,X_iX_j, where theXiX_i’s form a stratified system of Hörmander vector fields andai,ja_{i,j}are Hölder continuous functions belonging to a suitable class of ellipticity.

Let Ω be a domain in ℝ N and consider a second order linear partial differential operator A in divergence form on Ω which is not required to be uniformly elliptic and whose coefficients are allowed to be complex, unbounded and measurable. Under rather general conditions on the growth of the coefficients we construct a quasi-contractive analytic semigroup \((e^{-t A_{V}})_{t\geqslant0}\) on L 2(Ω,dx), whose generator A V gives an operator realization of A under general boundary conditions. Under suitable additional conditions on the imaginary parts of the diffusion coefficients, we prove that for a wide class of boundary conditions, the semigroup \((e^{-t A_{V}})_{t\geqslant0}\) is quasi-L p -contractive for 1<p<∞. Similar results hold for second order nondivergence form operators whose coefficients satisfy conditions similar to those on the coefficients of the operator A, except for some further requirements on the diffusion coefficients. Some examples where our results can be applied are provided.

On the atomic decomposition for Hardy spaces on Lipschitz domains of [formula omitted

We consider a boundary value problem for an elliptic differential operator of order 2m in a domain \({\mathcal{D}} \subset {\mathbb{R}}^{n}\). The boundary of \({\mathcal{D}}\) is smooth outside a smooth manifold Y of dimension 0 ≤ q < n − 1, and \(\partial {\mathcal{D}}\) bears edge type singularities along Y . The Lopatinskii condition is assumed to be fulfilled on the smooth part of \(\partial {\mathcal{D}}\). The corresponding spaces are weighted Sobolev spaces \(H^{{s,\gamma }} {\left( {\mathcal{D}} \right)}\), and this allows one to define ellipticity of weight γ for the problem. The resolvent of the problem is assumed to possess rays of minimal growth. The main result says that if there are rays of minimal growth with angles between neighbouring rays not exceeding π(γ + 2m)/n, then the root functions of the problem are complete in \(L^{2} ({\mathcal{D}})\). In the case of second order elliptic equations the results remain true for all domains with Lipschitz boundary.

The method of layer potentials has been applied with tremendous success in the treatment of boundary value problems for second-order differential operators for a very long time; the literature on this topic is enormous. By way of contrast this method is disproportionally underdeveloped in the case of higher-order operators; the difference between the higher-order and the second-order settings is striking in term of the scientific output. This paper presents new results which establish mapping properties of multiple layer potentials associated with higher-order elliptic operators in Lipschitz domains in ℝ n .

We consider the quasilinear diffusion problem of for an open set , , for , and any . Here, denotes an operator which may involve the distributional Riesz fractional gradient of order , with , the classical gradient or/and nonlocal derivatives , with . We show global existence results for various quasilinear diffusion systems in nondivergence form for linear elliptic operators , including classical elliptic systems, anisotropic fractional equations and systems, and anisotropic local and nonlocal operators of the following type: for coercive, invertible matrices and suitable vectorial functions .