In this paper, we investigate various comparison principles for quasilinear elliptic equations of Formula: see text-Laplace type with lower-order terms that depend on the solution and its gradient. ...More specifically, we study comparison principles for equations of the following form: −Δpu + H(u,Du) = 0,x ∈ Ω, where Formula: see text is the Formula: see text-Laplace operator with Formula: see text, and Formula: see text is a continuous function that satisfies a structure condition. Many of these results lead to comparison principles for the model equations Δpu = f(u) + g(u)|Du|q,x ∈ Ω, where Formula: see text are non-decreasing and Formula: see text. Our results either improve or complement those that appear in the literature.
In this paper we study minimal realizations in Lp(RN) of the second order elliptic operatorAb,c:=(1+|x|α)Δ+b|x|α−2x⋅∇−c|x|α−2−|x|β,x∈RN, where N≥3, α∈0,2), β>0, and b,c are real numbers. We use ...quadratic form methods to prove that (Ab,c,Cc∞(RN∖{0})) admits an extension that generates an analytic C0-semigroup for all p∈(1,∞). Moreover, we give conditions on the coefficients under which this extension is precisely the closure of (Ab,c,Cc∞(RN∖{0})).
The theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic differential operator on Rn with linear boundary conditions on ...(a relatively open part of) a compact hypersurface. Our approach allows to obtain Kreĭn-like resolvent formulae where the reference operator coincides with the “free” operator with domain H2(Rn); this provides an useful tool for the scattering problem from a hypersurface. Concrete examples of this construction are developed in connection with the standard boundary conditions, Dirichlet, Neumann, Robin, δ and δ′-type, assigned either on a (n−1) dimensional compact boundary Γ=∂Ω or on a relatively open part Σ⊂Γ. Schatten–von Neumann estimates for the difference of the powers of resolvents of the free and the perturbed operators are also proven; these give existence and completeness of the wave operators of the associated scattering systems.
The present paper concerns divergence form elliptic and degenerate elliptic operators in a domain Ω⊂Rn, and establishes the equivalence between the uniform rectifiability of the boundary E=∂Ω and ...weak Carleson condition on the good approximation of the Green function G by affine, or distance, functions. There are two main original contexts for the results, elliptic operators in a non-tangential access domain with an n−1 dimensional boundary and degenerate elliptic operators adapted to a domain with an Ahlfors regular boundary of larger co-dimension. In both cases necessary and sufficient conditions are given, in the form of Carleson packing conditions on the collection of balls centered on E where G is not well approximated.(1)This is the first time the underlying property of the control of the Green function by affine functions, or by the distance to the boundary, in the sense of the Carleson prevalent sets, appears in the literature; some results established here are new even in the half space;(2)the results are optimal, providing a full characterization of uniform rectifiability under the (standard) mild topological assumptions;(3)to the best of the authors' knowledge, even in traditional domains with (n−1)-dimensional boundaries, this is the first free boundary result applying to all elliptic operators, without any restriction on the coefficients (the direct one assumes the standard, and necessary, Carleson measure condition);(4)this is the first free boundary result in higher co-dimensional setting and as such, the first PDE characterization of uniform rectifiability for a set of dimension d, d<n−1, in Rn. The paper offers a general way to deal with related issues considerably beyond the scope of the aforementioned theorem, including the question of approximability of the gradient of the Green function, and the comparison of the Green function to a certain version of the distance to the original set rather than distance to the hyperplanes.
We prove local boundedness, Harnack inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form. Degeneracy is via a non negative, symmetric, ...measurable matrix-valued function Q(x) and two suitable non negative weight functions. We setup an axiomatic approach in terms of suitable geometric conditions and local Sobolev–Poincaré inequalities. Data integrability is close to L1 and it is exploited in terms of suitable version of Stummel-Kato class that in some cases is also necessary to the regularity.
Brownian motion with general drift Kinzebulatov, D.; Semënov, Yu. A.
Stochastic processes and their applications,
20/May , Letnik:
130, Številka:
5
Journal Article
Recenzirano
Odprti dostop
We construct and study the weak solution to stochastic differential equation dX(t)=−b(X(t))dt+2dW(t), X(0)=x, for every x∈Rd, d≥3, with b in the class of weakly form-bounded vector fields, ...containing, as proper subclasses, a sub-critical class Ld+L∞d, as well as critical classes such as weak Ld class, Kato class, Campanato–Morrey class.
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