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On the strong uniqueness of some finite dimensional Dirichlet operators
Abstract
We prove essential self-adjointness of a class of Dirichlet operators in ℝn using the hyperbolic equation approach. This method allows one to prove essential self-adjointness under minimal conditions on the logarithmic derivative of the density and a condition of Muckenhoupt type on the density itself.
... If d ≥ 3 and ρ is locally bounded and locally uniformly positive and there exists R > 0 such that ∇ρ ∈ L d loc (R d \ B R ) ∩ L 4 loc (R d ), then (β) follows by the Hölder and Sobolev inequalities. Therefore, [15,Theorem 1] can be regarded as a generalization of [2,Theorem 7] in case d ≥ 4. On the other hand, M. Kolb developed in [12,Theorem 1] another criterion for essential selfadjointness of (L, C ∞ 0 (R d )), which is stated as follows. If the condition (γ) below holds, then ...
... Remarkably, regardless of dimension d ≥ 1, if ρ is locally bounded and locally unifomly positive and ∇ρ ∈ L 4 loc (R d ), then the condition (γ) is fulfilled, so that (L, C ∞ 0 (R d )) is essentially self-adjoint by [12,Theorem 1]. Thus, [12,Theorem 1] is more general than [15,Theorem 1], if ρ is locally bounded and locally unifomly positive. ...
... Remarkably, regardless of dimension d ≥ 1, if ρ is locally bounded and locally unifomly positive and ∇ρ ∈ L 4 loc (R d ), then the condition (γ) is fulfilled, so that (L, C ∞ 0 (R d )) is essentially self-adjoint by [12,Theorem 1]. Thus, [12,Theorem 1] is more general than [15,Theorem 1], if ρ is locally bounded and locally unifomly positive. However, if d = 2 or 3, then the condition that ρ is locally bounded and locally unifomly positive and that ∇ρ ∈ L 4 loc (R d ) is still more ristrictive than (α ′ ). ...
We show the -uniqueness for any and the essential self-adjointness of a Dirichlet operator , with and . In particular, is allowed to be in or in for some , while is required to be locally bounded below and above by strictly positive constants. The main tools in this paper are elliptic regularity results for divergence and non-divergence type operators and basic properties of Dirichlet forms and their resolvents.
We show [Formula: see text]-uniqueness for any [Formula: see text] and the essential self-adjointness of a Dirichlet operator [Formula: see text], [Formula: see text] with [Formula: see text] and [Formula: see text]. In particular, [Formula: see text] is allowed to be in [Formula: see text] or in [Formula: see text] for some [Formula: see text], while [Formula: see text] is required to be locally bounded below and above by strictly positive constants. The main tools in this paper are elliptic regularity results for divergence and non-divergence type operators and basic properties of Dirichlet forms and their resolvents.
In this paper we extend the well-known Leinfelder–Simader theorem on the essential selfadjointness of singular Schrödinger operators to arbitrary complete Riemannian manifolds. This improves some earlier results of Shubin, Milatovic and others.
35J15 General theory of second-order, elliptic equations
35K10 General theory of second-order, parabolic equations
35R05 PDE with discontinuous coefficients or data
35R15 Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) (See also 46Gxx, 58D25)
58J65 Diffusion processes and stochastic analysis on manifolds (See also 35R60, 60H10, 60J60)
60J60 Diffusion processes (See also 58J65)
We obtain several essential self-adjointness conditions for a Schroedinger
type operator D*D+V acting in sections of a vector bundle over a manifold M.
Here V is a locally square-integrable bundle map. Our conditions are expressed
in terms of completeness of certain metrics on M; these metrics are naturally
associated to the operator. We do not assume a priori that M is endowed with a
complete Riemannian metric. This allows us to treat e.g. operators acting on
bounded domains in the euclidean space.
For the case when the principal symbol of the operator is scalar, we
establish more precise results. The proofs are based on an extension of the
Kato inequality which modifies and improves a result of Hess, Schrader and
Uhlenbrock.
Consider a symmetric bilinear form ϕdefined on ∞c(d) by[formula]In this paper we study the stochastic process associated with the smallest closed markovian extension of (ϕ, ∞c), and give a new proof of Markov uniqueness (i.e. the uniqueness of a closed markovian extension) based on purely probabilistic arguments. We also give another purely analytic one. As a consequence, we show that all invariant measures are reversible, provided they are of finite energy. The problem of uniqueness of such measures is also partially solved.
Two uniqueness results for C0 semigroups on weighted Lp spaces over Rn generated by operators of type Δ+β·∇ with singular drift β are proven. A key ingredient in the proofs is the verification of some kind of “weak Kato inequality” which seems to break down exactly for those drift singularities where Lp uniqueness breaks down as well.
We study the equation , where H is the operator associated with the Dirichlet form of quasi-invariant measure on Hilbert space. A priori estimates of the first and second order derivatives of the solutions of the equation are obtained. Then these estimates are applied to the problem of essential self-adjointness of the operator H on domains which consist of smooth functions.
We prove new pointwise inequalities involving the gradient of a function
u Î C1 ( \mathbbRn )u \in C^1 \left( {\mathbb{R}^n } \right)
, the modulus of continuity
w\omega
of the gradient
Ñu\nabla u
, and a certain maximal function
M¨ u\mathcal{M}^\diamondsuit u
and show that these inequalities are sharp. A simple particular case corresponding to
n = 1n = 1
and
w( r ) = r\omega \left( r \right) = r
is the Landau type inequality
| u¢( x ) |2 \leqslant \frac83M¨ u( x )M¨ u"( x )\left| {u'\left( x \right)} \right|^2 \leqslant \frac{8}{3}\mathcal{M}^\diamondsuit u\left( x \right)\mathcal{M}^\diamondsuit u''\left( x \right)
, where the constant 8/3 is best possible and
\mathcal{M}^\diamondsuit u''\left( x \right) = \mathop {\sup }\limits_{r > 0} \frac{1}{{2r}}\left| {\int_{x - r}^{x + r} {{\text{sign}}\left( {y - x} \right)u\left( y \right)dy} } \right|
.
Let ℒ≔Δ/2+(∇φ/φ) ·∇ be a generalized Schrödinger operator or generator of Nelsons diffusion, defined on C
∞
0(D) where φ is a continuous and strictly positive function on an open domain D⊂ℝ
d
such that ∇φ∈L
loc
2(D). Some results are given about the two questions below: (i) Whether does ℒ generate a unique semigroup in L
1(D, φ2
dx)? (ii) Whether the semigroup determined by ℒ is strong Feller?
Strong uniqueness inL2and inL1for Dirichlet operators in finite dimensional spaces is studied.
Using the theory of hyperbolic equations, simple conditions are given which ensure the essential self-adjointness of all powers of certain formally symmetric differential operators. Typical applications include Dirac and Laplace-Beltrami operators on complete Riemannian manifolds, as well as semibounded operators of Schrödinger type.
We prove a new regularity result for operators of type L = Delta +BDeltar+c, on open setsOmega ae IR n , provided B :Omega ! IR n , c :Omega ! IR satisfy mild integrability conditions. As a consequence we prove that L = Delta + r log ae Delta r with domain C 1 0 (IR n ) is essentially self-adjoint on L 2 (IR n ; aedx), if ae 2 H 1;1 loc (IR n ) and r log ae 2 L fl loc (IR n ; dx) for some fl ? n. AMS Subject Classification Primary: 47 B 25, 35 B 65 Secondary: 31 C 25 Key words and phrases: Dirichlet operators, generalized Schrodinger operators, regularity of weak solutions, essential self-adjointness Running head: Dirichlet operators on IR n 1) Department of Mechanics and Mathematics, Moscow State University, 119899 Moscow, Russia 2) School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA 3) Fakultat fur Mathematik, Universitat Bielefeld, D--33501 Bielefeld, Germany 1 One of the classical problems in mathematical physics is th...
Starting from the example of solving the Schrodinger equation, the concept of (essential) self-adjointness of a linear operator in a Hilbert space is developed. Some general criteria to prove this quality are presented and applied to Schrodinger operators of the form Gamma4 + V in L 2 Gamma R d Delta . Keywords: Schrodinger equation, Hilbert space, self-adjointness, Schrodinger operator 0
Analysis on local Dirichlet spaces II
- K T Sturm
K. T. Sturm, Analysis on local Dirichlet spaces II, Osaka J. Math. 32 (1996) 275–312.