Partial Differential Equations with Minimal Smoothness and Applications

This note is intended to be the first in a series of papers treating linear elliptic systems of partial differential operators subject to obstacle type constraints. There is a large literature concerning solutions to linear and nonlinear elliptic systems of partial differential equations, but there seems to be much less work devoted exclusively to the understanding of solutions to such systems when they are subject to constraints. These constrained systems often take the form of a system of variational inequalities. Such problems have been treated, for example in [F1] and [HW] at least for strongly elliptic, i.e. Legendre-Hadamard elliptic, systems of variational inequalities. In this note we want to begin a study of a broader class of such systems, what we shall refer to as weakly elliptic systems — elliptic in the standard sense that the characteristic form of the principal part has no real zeros. One very important feature of this larger class is that the solution vector(s) can not in general, have the same degree of regularity in each component direction, as is generally the case for strongly elliptic systems. And as is generally well understood, solutions to variational inequalities only inherit a very limited amount of regularity from the data. For weakly elliptic linear systems, this inherited regularity is intimately tied up with certain algebraic structure considerations, considerations that do not appear when the obstacle constraints are removed.

An elementary proof is given of Widder’s Theorem and of the uniqueness of isolated singularities for parabolic differential equations Lu = u _t. It applies equally well to operators L that are Holder continuous, in divergence form, or of Hörmander type.

Generalized differential operators are ones which agree with differential operators when applied to sufficiently smooth functions but have special symmetry properties which allow them to be defined on less smooth functions. Such operators were used by Cantor [4] in his proof of the uniqueness of representation by trigonometric series and have been an integral part of all extensions of Cantor’s theorem to higher dimensions.

In a Lipschitz domain Ω ⊂ R ⁿ, associated to the p-Laplace equation one can define a notion of p-harmonic measure on subsets E ⊂ ∂Ω by solving the Dirichlet problem for (1) with boundary values χE. Denote by w _p(x; E) the p-harmbnic function with boundary values 1 on E and 0 on ∂Ω\E. In the linear case p = 2 for each x ∈ Ω we do obtain a Borei measure on ∂Ω. This is no longer true in the nonlinear case p ≠ 2. Yet the monotonicity properties of classical harmonic measures extend to their nonlinear counterparts. Many applications of this principle are in [GLM], [HM].

Let D be a domain in R ⁿ, n ≥ 2, and let B _t be Brownian motion in D with lifetime τD. The transition probabilities for this motion are given by the Dirichlet heat kernel P _t ^D(x, y) for \(\frac{1}{2}\Delta\) in D. If h is a positive harmonic function in D the Doob h-process, the Brownian motion conditioned by h, is determined by the following transition functions: .

Let us consider the class of elliptic equations defined on a bounded C ² domain D in R ⁿ.

The purpose of this note is to discuss some conjectures concerning harmonic measure. We will start by considering harmonic measure on a simply connected plane domain Ω, but eventually we will also consider some multiply connected and higher dimensional domains. Most of these questions are trivial if ∂Ω has tangents a.e. and many are easy if Ω is only a quasicircle. Thus they are really questions about very non-smooth domains.

In this note, I would like to describe some recent work that considers the relationship between the smoothness of the boundary of a domain in R ⁿ and the spectral properties of the Laplacian in the domain.

Let Ω be a bounded Lipschitz domain in R ⁿ. Consider the parabolic system where 0<T<∞, μ > 0 and λ < −2μ/n are constants.

Let \( p_t^D(x,y) = p_t(x,y) \) be the Dirichlet heat kernel for \( \frac{1}{2}\Delta \) in a domain D ⊂ R ⁿ, n ≥ 2. In [4] E.B. Davies and B. Simon define the semigroup connected with the Dirichlet Laplacian to be intrinsically ultracontractive if there is a positive (in D) eigenfunction φ ₀ for \( \frac{1}{2}\Delta \) in D and if for each t > 0 there are positive constants c _t, C_t depending only on D and t such that .

Uniqueness is proved for the Dirichlet problem associated to a second order non-divergence form elliptic operator whose coefficients are independent of one direction and have discontinuities along a countable set of lines which are parallel to the given direction and accumulate around a single line.

Let D denote a bounded Lipschitz domain in R ⁿ. For almost every (with respect to surface measure dσ)Q ∈∂D the exterior normal N _Q at Q exists. The solution u to the Dirichlet problem, , with g ∈ L ²(∂D,dσ) can be represented in the form of the classical double layer potential .

A famous result, first proved in ℝ² by Carleman [C] in 1939, states that if \(V \in L_{\text{loc}}^{\infty}(\mathbb{R}^N)\) and u is a solution to Δu = Vu in a connected open set \(D \subset \mathbb{R}^N\), then u cannot vanish to infinite order at a point x ₀ ∈ D unless u ≡ 0 in D. We are interested in analogous results when the (elliptic) Laplacian in ℝ^N is replaced by a subelliptic operator of the type , where X ₁,…, X _{N −1} are smooth vector fields satisfying Hörmander’s condition for hypoellipticity [H]: .

In this note we will prove estimates for harmonic measure in convex and convex C ¹ domains. It is not hard to show that in a convex domain, surface measure belongs to the Muckenhoupt class A ₁ with respect to harmonic measure (Lemma 3). If the boundary of the domain is also of class C ¹, then it follows from [JK1] that the constant in the A ₁ condition tends to 1 as the radius of the ball tends to 0 (Lemma 7′). Our main estimates (Theorems A and B) are of the same type. The novelty is that they are not calculated with respect to balls, but rather with respect to “slices” formed by the intersection of the boundary with an arbitrary half-space. In addition to proving Theorems A and B we will also explain how these estimates are related to an approach to regularity for the Monge-Ampère equation due to L. Caffarelli and to a problem of prescribing harmonic measure as a function of the unit normal.

Let Ω be a smooth, bounded and connected domain in R ⁿ. In this paper, we consider the following boundary value problem: .

Let ℝ be the real numbers and if E \( \subseteq \) ℝ, let Ē,∂E, |E|, denote the closure, boundary, and outer Lebesgue measure of E, respectively.

Let X _t be a continuous martingale starting at 0 and set \( X^\ast = \text{sup}\{\left\vert X_t \right\vert : t > 0 \} \) and \( S(X) = \sqrt{\left\langle X \right\rangle_\infty} \), where 〈X〉_tis the quadratic variation process at time t. We then define a measure μ on ℝ by \( \mu(E) = d\left\langle X \right\rangle_t (\{t : X_t \in E\}) \), where \( d\left\langle X \right\rangle_t \) is the Riemann-Stieltjes measure on 0,∞) associated to the nondecreasing function 〈X〉_t. It is known that μ is absolutely continuous with respect to Lebesgue measure so that there exists a function L(a), (called the local time), so that \( \mu(E) = \int_E L(a)da \) for every Borel set E. More generally, for any Borel function f on ℝ: Take f = 1 in (1.1) and we obtain: where the last equality follows by noting that \( L(a) = 0 \text{ if} a \notin [-X^\ast, X^\ast] \). We now set \( L^\ast = \text{sup}\{L(a) : a \in \mathbb{R}\} \); L* is called the maximal local time. Then (1.2) trivially yields \( S(X)^2\leq 2L^\ast X^\ast \), and this, the Cauchy-Schwarz inequality, and the Burkholder-Gundy inequality: \( \left \Vert S(X) \right \Vert_p\approx \left \Vert X^\ast \right \Vert_p for\ 0< p< \infty \), then gives \( \left \Vert S(X) \right \Vert_p\leq C_p\left \Vert L\ast \right \Vert_p \text{ for} 0 < p < \infty \). The reverse inequality is more difficult and was shown by Barlow and Yor [4], [5]. In fact, even more is true. The following theorem is essentially to Bass [6] and independently, Davis [12], however, their statements of these results are different than appears here, but a careful analysis of their methods yields these results.

Restriction theorems for Fourier integrals and series are proven. In the continuous case, a mixed norm inequality for the restriction of the Fourier transform to the sphere is given. On the other hand a discrete version of a result by R.S. Strichartz [10] is found.

In a partial differential equation roughness in the solution may be caused in various ways, such as by

Example A) In a linear Poisson problem, roughness introduced by i) or ii) leads to dramatically different behavior in numerical solutions.

Example B) Although for linear and semilinear parabolic problems, in the case of iii) (all other “data” being smooth) there is “no difference” in the sense that the solution is smooth for (some) positive time, in numerical analysis the linear and semilinear problem are miles apart.

While the main point of Example A was “computational folklore” for many years before it was fully proven in 1984, the punchline of Example B was totally unexpected at the time of its discovery in 1987.

More details, and more examples of rough stuff in the numerical analysis of partial differential equations, e.g., in singularly perturbed problems, can be found in WAHLBIN [7].