MODULUS OF CONTINUITY OF THE DIRICHLET SOLUTIONS

Let D be a bounded domain in Rn with n >= 2. For a function f on ∂D we denote by HDf the Dirichlet solution of f over D. It is classical that if D is regular, then HD maps the family of continuous boundary functions to the family of harmonic functions in D continuous up to the boundary ∂D. We show that the better continuity of a boundary function f ensures the better continuity of HDf in the context of general modulus of continuity

Equivalence between the boundary Harnack principle and the Carleson estimate

Both the boundary Harnack principle and the Carleson estimate describe the boundary behavior of positive harmonic functions vanishing on a portion of the boundary. These notions are inextricably related and have been obtained simultaneously for domains with specific geometrical conditions. The main aim of this paper is to show that the boundary Harnack principle and the Carleson estimate are equivalent for arbitrary domains

DOUBLING CONDITIONS FOR HARMONIC MEASURE IN JOHN DOMAINS

We introduce new classes of domains, i.e., semi-uniform domains and inner emi-uniform domains. Both of them are intermediate between the class of John domains nd the class of uniform domains. Under the capacity density condition, we show that the armonic measure of a John domain D satisfies certain doubling conditions if and only if is a semi-uniform domain or an inner semi-uniform domain

Hölder estimates of p-harmonic extension operators

AbstractIt is now a well-known fact that for 1<p<∞ the p-harmonic functions on domains in metric measure spaces equipped with a doubling measure supporting a (1,p)-Poincaré inequality are locally Hölder continuous. In this note we provide a characterization of domains in such metric spaces for which p-harmonic extensions of Hölder continuous boundary data are globally Hölder continuous. We also provide a link between this regularity property of the domain and the uniform p-fatness of the complement of the domain

On boundary layers

https://projecteuclid.org/download/pdf_1/euclid.afm/148589849

Intrinsic ultracontractivity for domains in negatively curved manifolds

Let $M$ be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected sets $D$ in $M$ for which the corresponding Dirichlet heat semigroup is intrinsically ultracontractive. That condition is formulated in terms of capacitary width. It is shown that both the reciprocal of the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(D)$, and the supremum of the torsion function for $D$ are comparable with the square of the capacitary width for $D$ if the latter is sufficiently small. The technical key ingredients are the volume doubling property, the Poincar\'e inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel for finite scale.Comment: 21 pages, 2 figure

The 3G inequality for a uniformly John domain

Let G be the Green function for a domain D $\subset$ Rd with d ≥ 3. The Martin boundary of D and the 3G inequality:$\frac{G(x,y)G(y,z)}{G(x,z)} \le A(|x-y|^{2-d}+|y-z|^{2-d})$ for x,y,z $\in$ Dare studied. We give the 3G inequality for a bounded uniformly John domain D, although the Martin boundary of D need not coincide with the Euclidean boundary. On the other hand, we construct a bounded domain such that the Martin boundary coincides with the Euclidean boundary and yet the 3G inequality does not hold

The 3G inequality for a uniformly John domain

Let G be the Green function for a domain D $\subset$ Rd with d ≥ 3. The Martin boundary of D and the 3G inequality:$\frac{G(x,y)G(y,z)}{G(x,z)} \le A(|x-y|^{2-d}+|y-z|^{2-d})$ for x,y,z $\in$ Dare studied. We give the 3G inequality for a bounded uniformly John domain D, although the Martin boundary of D need not coincide with the Euclidean boundary. On the other hand, we construct a bounded domain such that the Martin boundary coincides with the Euclidean boundary and yet the 3G inequality does not hold

Intra-articular osteoid osteoma of the calcaneus: a case report and review

AbstractOsteoid osteoma of the calcaneus is rare and frequently misdiagnosed as arthritis because of similar symptoms. In addition, radiographic findings may be nonspecific, and magnetic resonance imaging (MRI) may show a bone marrow edema and changes in adjacent soft tissue. A 19-year-old man presented with a 6-month history of persistent pain and swelling in the left hind foot; diagnostic computed tomography and MRI analyses revealed lesions suggesting an intra-articular osteoid osteoma of the calcaneus. Initial MRI did not show specific findings. On operation, the tumor was removed by curettage; pathologic findings demonstrated woven bone trabeculae surrounded by connective tissue, confirming the diagnosis. To the best of our knowledge, MRI scans in all cases of calcaneal osteoid osteoma reported till 3 months after the injury exhibited a nidus. We believe that calcaneal osteoid osteoma should be considered as a differential diagnosis in patients undergoing MRI 3 months after symptom presentation; early computed tomography is critical in diagnosis