2013/12/12 by Ross G. Pinsky, Ross Pinsky, Pinsky, Ross
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP
paper · pdf · doi:10.48550/arxiv.1312.3432
arxiv created 2013/12/12 · openalex publication_date 2013/12/12 · arxiv updated 2013/12/13 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
Fix R>0 and let BR denote the ball of radius R centered at the origin in Rd, d≥2. Let D⊂ BR be an open set with smooth boundary and such that Rd- D is connected, and let L=∑i,j=1dai,j(∂2)/(∂ xi∂ xj)-∑i=1dbi(∂)/(∂ xi) be a second order elliptic operator. Consider the following linear heat equation in the exterior domain Rd- D with boundary flux: \beginaligned L u=0 in Rd- D;
a∇ u⋅ n=-h on ∂ D;
u>0 is minimal, \endaligned where h\gneq0 is continuous, and where n is the unit inward normal to the domain Rd- D. The operator L must possess a Green's function in order that a solution u exist. An important feature of the equation is that there is no a priori bound on the supremum supx∈ Rd- Du(x) of the solution exclusively in terms of the boundary flux h, the hyper-surface measure of ∂ D and the coefficients of L; rather the geometry of D⊂ BR plays an essential role. However, we prove that in the case that L is a \it symmetric operator\rm with respect to some reference measure, then \it outside of\rm BR, the solution to \eqrefLHE is uniformly bounded, independent of the particular choice of D⊂ BR. The proof uses a combination of analytic and probabilistic techniques.