| 1. |
- Auscher, Pascal, et al.
(författare)
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L2 well-posedness of boundary value problems and the Kato square root problem for parabolic systems with measurable coefficients
- 2020
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Ingår i: Journal of the European Mathematical Society (Print). - : European Mathematical Society Publishing House. - 1435-9855 .- 1435-9863. ; 22:9, s. 2943-3058
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a first order strategy to study boundary value problems of parabolic systems with second order elliptic part in the upper half-space. This involves a parabolic Dirac operator at the boundary. We allow for measurable time dependence and some transversal dependence in the coefficients. We obtain layer potential representations for solutions in some classes and prove new well-posedness and perturbation results. As a byproduct, we prove for the first time a Kato estimate for the square root of parabolic operators with time dependent coefficients. This considerably extends prior results obtained by one of us under time and transversal independence. A major difficulty compared to a similar treatment of elliptic equations is the presence of non-local fractional derivatives in time.
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| 2. |
- Azzam, Jonas, et al.
(författare)
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A new characterization of chord-arc domains
- 2017
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Ingår i: Journal of the European Mathematical Society (Print). - 1435-9855 .- 1435-9863. ; 19:4, s. 967-981
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Tidskriftsartikel (refereegranskat)abstract
- We show that if Ω⊂Rn+1, n≥1, is a uniform domain (also known as a 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of Ω implies the existence of exterior corkscrew points at all scales, so that in fact, Ω is a chord-arc domain, i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior corkscrew conditions, and an interior Harnack chain condition. We discuss some implications of this result for theorems of F. and M. Riesz type, and for certain free boundary problems.
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| 3. |
- Lewis, John L, et al.
(författare)
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On the dimension of p-harmonic measure in space
- 2013
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Ingår i: Journal of the European Mathematical Society (Print). - 1435-9855 .- 1435-9863. ; 15:6, s. 2197-2256
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Tidskriftsartikel (refereegranskat)abstract
- Let Omega subset of R-n, n >= 3, and let p, 1 < p < infinity, p not equal D 2, be given. In this paper we study the dimension of p-harmonic measures that arise from nonnegative solutions to the p-Laplace equation, vanishing on a portion of partial derivative Omega, in the setting of delta-Reifenberg flat domains. We prove, for p >= n, that there exists (delta) over tilde = (delta) over tilde (p, n) > 0 small such that if Omega is a delta-Reifenberg flat domain with delta < <(delta)over tilde>, then p-harmonic measure is concentrated on a set of sigma-finite Hn-1-measure. We prove, for p >= n, that for sufficiently flat Wolff snowflakes the Hausdorff dimension of p-harmonic measure is always less than n - 1. We also prove that if 2 < p < n, then there exist Wolff snowflakes such that the Hausdorff dimension of p-harmonic measure is less than n - 1, while if 1 < p < 2, then there exist Wolff snowflakes such that the Hausdorff dimension of p-harmonic measure is larger than n - 1. Furthermore, perturbing off the case p = 2; we derive estimates for the Hausdorff dimension of p-harmonic measure when p is near 2.
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| 4. |
- Lewis, John L., et al.
(författare)
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Quasi-linear PDEs and low-dimensional sets
- 2018
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Ingår i: Journal of the European Mathematical Society (Print). - 1435-9855 .- 1435-9863. ; 20:7, s. 1689-1746
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we establish new results concerning boundary Harnack inequalities and the Martin boundary problem, for non-negative solutions to equations of $p$-Laplace type with variable coefficients. The key novelty is that we consider solutions which vanish only on a low-dimensional set $\Sigma$ in $\mathbb R^n$ and this is different compared to the more traditional setting of boundary value problems set in the geometrical situation of a bounded domain in $\mathbb R^n$ having a boundary with (Hausdorff) dimension in the range $[n-1,n)$. We establish our quantitative and scale-invariant estimates in the context of low-dimensional Reifenberg flat sets.
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