Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups

Let X0,X1,…,Xq(q<N) be real vector fields, which are left invariant on homogeneous group G, provided that X0 is homogeneous of degree two and X1,…,Xq are homogeneous of degree one. We consider the following nondivergence degenerate operator with drift L=∑i,j=1qaij(x)XiXj+a0(x)X0, where the coefficients aij(x), a0(x) belonging to vanishing mean oscillation space are bounded measurable functions. Furthermore, aij(x) satisfies the uniform ellipticity condition on Rq and a0(x)≠0. We obtain the local weighted Sobolev–Morrey estimates by applying the boundedness of commutators and interpolation inequalities on weighted Morrey spaces.

Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity

<p style='text-indent:20px;'>We determine the asymptotic spreading speed of the solutions of a Fisher-KPP reaction-diffusion equation, starting from compactly supported initial data, when the diffusion coefficient is a fixed bounded monotone profile that is shifted at a given forcing speed and satisfies a general uniform ellipticity condition. Depending on the monotonicity of the profile, we are able to characterize this spreading speed as a function of the forcing speed and the two linear spreading speeds associated to the asymptotic problems at <inline-formula><tex-math id="M1">\begin{document}$ x = \pm \infty $\end{document}</tex-math></inline-formula>. Most notably, when the profile of the diffusion coefficient is increasing we show that there is an intermediate range for the forcing speed where spreading actually occurs at a speed which is larger than the linear speed associated with the homogeneous state around the position of the front. We complement our study with the construction of strictly monotone traveling front solutions with strong exponential decay near the unstable state when the profile of the diffusion coefficient is decreasing and in the regime where the forcing speed is precisely the selected spreading speed.</p>

A note on positivity of two-dimensional differential operators

We consider the two-dimensional differential operator A(t,x)u(t,x) = -a11 (t, x) utt -a22(t,x)uxx +?u defined on functions on the half-plane R+ x R with the boundary condition u(0,x) = 0, x ? R where aii(t,x), i = 1,2 are continuously differentiable and satisfy the uniform ellipticity condition a2 11(t,x) + a222(t,x)? ? > 0, ? > 0. The structure of fractional spaces E?,1 (L1 (R+ x R), A(t,x)) generated by the operator A(t,x) is investigated. The positivity of A(t,x) in L1 (W2?1(R+ x R)) spaces is established. In applications, theorems on well-posedness in L1 (W2?1 (R+ x R)) spaces of elliptic problems are obtained.

Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients

We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the affine form a = a(y) = ā + ∑j≥1yjψj for some parameter vector y = (yj)j ≥ 1 ∈ U = [−1,1]N. We study the summability properties of polynomial expansions of the solution map y → u(y) ∈ V := H01(D) . We consider both Taylor series and Legendre series. Previous results [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47] show that, under a uniform ellipticity assuption, for any 0 <p< 1, the ℓp summability of the (∥ψj∥L∞)j ≥ 1 implies the ℓp summability of the V-norms of the Taylor or Legendre coefficients. Such results ensure convergence rates n− s of polynomial approximations obtained by best n-term truncation of such series, with s = (1/p)−1 in L∞(U,V) or s = (1/p)−(1/2) in L2(U,V). In this paper we considerably improve these results by providing sufficient conditions of ℓp summability of the coefficient V-norm sequences expressed in terms of the pointwise summability properties of the (|ψj|)j ≥ 1. The approach in the present paper strongly differs from that of [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47], which is based on individual estimates of the coefficient norms obtained by the Cauchy formula applied to a holomorphic extension of the solution map. Here, we use weighted summability estimates, obtained by real-variable arguments. While the obtained results imply those of [7] as a particular case, they lead to a refined analysis which takes into account the amount of overlap between the supports of the ψj. For instance, in the case of disjoint supports, these results imply that for all 0 <p< 2, the ℓp summability of the coefficient V-norm sequences follows from the weaker assumption that (∥ψj∥L∞)j ≥ 1 is ℓq summable for q = q(p) := 2p/(2−p) . We provide a simple analytic example showing that this result is in general optimal and illustrate our findings by numerical experiments. The analysis in the present paper applies to other types of linear PDEs with similar affine parametrization of the coefficients, and to more general Jacobi polynomial expansions.

The Cauchy-Dirichlet Problem for a Class of Linear Parabolic Differential Equations with Unbounded Coefficients in an Unbounded Domain

We consider the Cauchy-Dirichlet problem in [0,∞)×D for a class of linear parabolic partial differential equations. We assume that D⊂ℝd is an unbounded, open, connected set with regular boundary. Our hypotheses are unbounded and locally Lipschitz coefficients, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution to the nonhomogeneous Cauchy-Dirichlet problem using stochastic differential equations and parabolic differential equations in bounded domains.