Newtonian potential theory for unbounded sources and applications to free boundary problems

Karp, Lavi; Margulis, Avmir S.

doi:10.1007/bf02820440

Cited by 40 publications

(51 citation statements)

References 37 publications

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“…For the ordinary analytic continuation this follows from the results in [6], or in [15]. The second property implies that a domain cannot have an outward cusp which does not satisfy (1.3), if χ has a multivalued analytic continuation.…”

Section: Introduction and The Main Resultsmentioning

confidence: 83%

Multivalued Analytic Continuation of the Cauchy Transform

Karp

2006

Potential Anal

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In this paper we introduce the notion of multivalued analytic continuation of the Cauchy transforms. Many difficulties arise because the continuation is not single-valued. Our main result asserts that if χ has a multivalued analytic continuation, then the free boundary ∂ has zero Lebesgue measure. Here χ is the characteristic function of a domain and ∂ is its boundary. We also discuss the connections between this notion, quadrature domains and approximations of analytic functions with single-valued integrals by rational functions. The last problem is related to the existence of a continuous function g and a closed connected set K such that the gradient of g vanishes on K, nevertheless g is not constant on K. (2000): Primary 31A25, 31B20; secondary 30E10, 35J05, 41A20. Mathematics Subject Classifications

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Section: Introduction and The Main Resultsmentioning

confidence: 83%

Multivalued Analytic Continuation of the Cauchy Transform

Karp

2006

Potential Anal

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“…The results of [17] apply in the situation that σ ≥ 1 everywhere, but we will want to relax this condition. We show only the minimal amount of regularity required for the proofs of our main theorems.…”

Section: Boundary Regularity For the Obstacle Problemmentioning

confidence: 99%

Scaling limits for internal aggregation models with multiple sources

Levine

Peres

2010

JAMA

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We study the scaling limits of three different aggregation models on Z d : internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R d . In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the DiaconisFulton smash sum of domains.

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“…In particular, they are allowed to be of positive measure. However, it is known (see, e.g., Lemma 2.11 in [13]) that when the first assumption of the theorem is satisfied then ∂G(t) and ∂H (t) have n-dimensional Lebesgue measure zero for t > 0.…”

Section: Weak Formulationmentioning

confidence: 99%

Nonbranching weak and starshaped strong solutions for Hele-Shaw dynamics

Gustafsson

Vasil'ev

2005

Arch. Math.

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We consider multidimensional weak and strong Hele-Shaw dynamics (t) of an advancing/receding viscous fluid injected/removed through a single finite point into/from a bounded domain (0). A class of weak solutions is shown to preserve local uniqueness in both directions. Then we also consider strong solutions (t), and show that if (0) is starshaped with respect to a small ball centered on the point of injection, then the evolution (t) exists for all time.

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Newtonian potential theory for unbounded sources and applications to free boundary problems

Cited by 40 publications

References 37 publications

Multivalued Analytic Continuation of the Cauchy Transform

Multivalued Analytic Continuation of the Cauchy Transform

Scaling limits for internal aggregation models with multiple sources

Nonbranching weak and starshaped strong solutions for Hele-Shaw dynamics

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