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

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 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} .

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 »

Let  ⇢ Rn be a bounded open set satisfying the uniform exterior cone condition. Let A be a uniformly elliptic operator given by Au = Xn i, j=1 ai j @i j u + Xn j=1 b j @ j u + cu where a j i = ai j 2 C(¯ ) and b j , c 2 L1(), c  0 . We show that the realization A0 of A in C0() := {u 2 C(¯ ) : u|@ = 0} given by D(A0) := {u 2 C0() W2,n loc () : Au 2 C0()} A0u := Au generates a bounded holomorphic C0-semigroup on C0(). The result is in particular true if  is a Lipschitz domain. So far the best known result seems to be the case where  has C2-boundary [12, Section 3.1.5]. We also study the elliptic problem −Au = f u|@ = g .

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 ...

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

We present a new method for the existence and pointwise estimates of a Green&amp;#39;s function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Green&amp;#39;s function for constant coefficients equations.

Parabolic Harnack inequality of viscosity solutions on Riemannian manifolds

Harnack inequality for a subelliptic PDE in nondivergence form

On certain degenerate and singular elliptic PDEs IV: Nondivergence-form operators with logarithmic degeneracies or singularities

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.

There is growing evidence that population-level brain activity is often organized into propagating waves that are structured in both space and time. Such spatiotemporal patterns have been linked to brain function and observed across multiple recording methodologies and scales. The ability to detect and analyze these patterns is thus essential for understanding the working mechanisms of neural circuits. Here we present a mathematical and computational framework for the identification and analysis of multiple classes of wave patterns in neural population-level recordings. By drawing a conceptual link between spatiotemporal patterns found in the brain and coherent structures such as vortices found in turbulent flows, we introduce velocity vector fields to characterize neural population activity. These vector fields are calculated for both phase and amplitude of oscillatory neural signals by adapting optical flow estimation methods from the field of computer vision. Based on these velocity vector fields, we then introduce order parameters and critical point analysis to detect and characterize a diverse range of propagating wave patterns, including planar waves, sources, sinks, spiral waves, and saddle patterns. We also introduce a novel vector field decomposition method that extracts the dominant spatiotemporal structures in a recording. This enables neural data to be represented by the activity of a small number of independent spatiotemporal modes, providing an alternative to existing dimensionality reduction techniques which separate space and time components. We demonstrate the capabilities of the framework and toolbox with simulated data, local field potentials from marmoset visual cortex and optical voltage recordings from whole mouse cortex, and we show that pattern dynamics are non-random and are modulated by the presence of visual stimuli. These methods are implemented in a MATLAB toolbox, which is freely available under an open-source licensing agreement.

In the paper one considers second-order elliptic equations with measurable coefficients in nondivergence form. One proves for them Harnack's inequality and one estimates the Holder exponent of the solutions. One makes no assumption regarding the smallness of the dispersion of the eigenvalues of the matrix of the coefficients of the second-order derivatives.

We study the three-dimensional magnetohydrodynamics system and obtain its regularity criteria in terms of only two velocity vector field components eliminating the condition on the third component completely. The proof consists of a new decomposition of the four nonlinear terms of the system and estimating a component of the magnetic vector field in terms of the same component of the velocity vector field. This result may be seen as a component reduction result of many previous works [C. He and Z. Xin, “On the regularity of weak solutions to the magnetohydrodynamic equations,” J. Differ. Equ. 213(2), 234–254 (2005); Y. Zhou, “Remarks on regularities for the 3D MHD equations,” Discrete Contin. Dyn. Syst. 12(5), 881–886 (2005)].