BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Bruno Poggi Cevallos (University of Minnesota) DTSTART:20200914T160000Z DTEND:20200914T165000Z DTSTAMP:20260423T010102Z UID:anpdews/33 DESCRIPTION:Title: Additive and scalar-multiplicative Carleson perturbations of elliptic ope rators on domains with low dimensional boundaries.\nby Bruno Poggi Cev allos (University of Minnesota) as part of HA-GMT-PDE Seminar\n\n\nAbstrac t\nAt the beginning of the 90s\, Fefferman\, Kenig and Pipher (FKP) obtain ed a rather sharp (additive) perturbation result for the Dirichlet problem of divergence form elliptic operators. Without delving into details\, the point is that if the (additive) disagreement of two operators satisfies w hat is known as a Carleson measure condition\, then quantitative absolute continuity of the elliptic measure is transferred from one operator to the other\, if one of the operators already possesses this property. Their (a dditive) perturbation result has since then been generalized to increasing ly weaker geometric and topological assumptions on boundaries of co-dimens ion 1\, by multiple authors. \n\nThis talk will consist of two main parts . In the first part\, we will see an extension of the FKP result to the d egenerate elliptic operators of David\, Feneuil and Mayboroda\, which were developed to study geometric and analytic properties of sets with boundar ies whose co-dimension is higher than 1. These operators are of the form - div A∇ \, where A is a degenerate elliptic matrix crafted to weigh the d istance to the low-dimensional boundary in a way that allows for the nouri shment of an elliptic theory. When this boundary is a d-Alhfors-David regu lar set in R^n with d in [1\, n-1)\, and n≥ 3\, we prove that the member ship of the elliptic measure in A_∞ is preserved under (additive) Carle son measure perturbations of the matrix of coefficients\, yielding in turn that the L^p-solvability of the Dirichlet problem is also stable under th ese perturbations (with possibly different p). If the Carleson measure pe rturbations are suitably small\, we establish solvability of the Dirichlet problem in the same L^p space. One of the corollaries of our results toge ther with a previous result of David\, Engelstein and Mayboroda\, is that\ , given any d-ADR boundary Γ with d in [1\, n-2)\, n≥ 3\, there is a f amily of degenerate operators of the form described above whose elliptic m easure is absolutely continuous with respect to the d-dimensional Hausdor ff measure on Γ. Our method of proof uses the method of Carleson measure extrapolation\, as developed by Lewis and Murray\, and adapted to a dyadi c setting by Hofmann and Martell in the past decade. This is joint work wi th Svitlana Mayboroda.\n\nIn the second part of the talk\, we will adopt a slightly different perspective than has been customary in the literature of these perturbation results\, by considering scalar-multiplicative Carle son perturbations\, as communicated to us by Joseph Feneuil and inspired b y the work on equations with drift terms of Hofmann and Lewis\, and Kenig and Pipher\, at the start of the 21st century. Essentially\, if we may wr ite A=bA_0 with b a scalar function bounded above and below by a positive number\, and ∇b·dist(· \,Γ) satisfying a Carleson measure condition\, then we still retain the transference of the quantitative absolute contin uity of the elliptic measure for -div A∇\, if -div A_0∇ already has th is property. By way of examples in the setting of low dimensional boundari es\, we will see that one ought to consider these two types of perturbatio ns (namely\, additive and scalar-multiplicative) to reckon a more complete picture of the absolute continuity of elliptic measure. This is joint wor k with Joseph Feneuil.\n LOCATION:https://researchseminars.org/talk/anpdews/33/ END:VEVENT END:VCALENDAR