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Wendelin Werner - One of the best experts on this subject based on the ideXlab platform.
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on bounded type thin local sets of the two dimensional gaussian free field
Journal of The Institute of Mathematics of Jussieu, 2017Co-Authors: Juhan Aru, Avelio Sepulveda, Wendelin WernerAbstract:We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a Simply Connected Domain, and their relation to the conformal loop ensemble $\text{CLE}_{4}$ and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the $\text{CLE}_{4}$ carpet, and prove that all BTLS are necessarily Connected to the boundary of the Domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of $\text{CLE}_{4}$ ) are in fact measurable functions of the GFF.
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on bounded type thin local sets of the two dimensional gaussian free field
arXiv: Probability, 2016Co-Authors: Juhan Aru, Avelio Sepulveda, Wendelin WernerAbstract:We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a Simply-Connected Domain, and their relation to the conformal loop ensemble CLE(4) and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded-type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the CLE(4) carpet, and prove that all BTLS are necessarily Connected to the boundary of the Domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of CLE(4)) are in fact measurable functions of the GFF.
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conformal invariance of planar loop erased random walks and uniform spanning trees
arXiv: Probability, 2001Co-Authors: Gregory F Lawler, Oded Schramm, Wendelin WernerAbstract:We prove that the scaling limit of loop-erased random walk in a Simply Connected Domain $D$ is equal to the radial SLE(2) path in $D$. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan Domain exists and is conformally invariant. Assuming that the boundary of the Domain is a $C^1$ simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc $A$ on the boundary, is the chordal SLE(8) path in the closure of $D$ joining the endpoints of $A$. A by-product of this result is that SLE(8) is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.
Salakhudinov R. - One of the best experts on this subject based on the ideXlab platform.
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Integral properties of the classical warping function of a Simply Connected Domain
2020Co-Authors: Salakhudinov R.Abstract:Let u(z,G) be the classical warping function of a Simply Connected Domain G. We prove that the L p-norms of the warping function with different exponents are related by a sharp isoperimetric inequality, including the functional u(G) = sup x∈Gu(x, G). A particular case of our result is the classical Payne inequality for the torsional rigidity of a Domain. On the basis of the warping function, we construct a new physical functional possessing the isoperimetric monotonicity property. For a class of integrals depending on the warping function, we also obtain a priori estimates in terms of the L p-norms of the warping function as well as the functional u(G). In the proof, we use the estimation technique on level lines proposed by Payne. © 2012 Pleiades Publishing, Ltd
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Isoperimetric monotony of the L p -norm of the warping function of a plane Simply Connected Domain
2020Co-Authors: Salakhudinov R.Abstract:Let G be a Simply Connected Domain and let u(x,G) be its warping function. We prove that L p -norms of functions u and u -1 are monotone with respect to the parameter p. This monotony also gives isoperimetric inequalities for norms that correspond to different values of the parameter p. The main result of this paper is a generalization of classical isoperimetric inequalities of St.Venant-Pólya and the Payne inequalities. © 2010 Allerton Press, Inc
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Isoperimetric properties of Euclidean boundary moments of a Simply Connected Domain
2020Co-Authors: Salakhudinov R.Abstract:We consider integral functionals of a Simply Connected Domain which depend on the distance to the Domain boundary. We prove an isoperimetric inequality generalizing theorems derived by the Schwarz symmetrization method. For L p -norms of the distance function we prove an analog of the Payne inequality for the torsional rigidity of the Domain. In compare with the Payne inequality we find new extremal Domains different from a disk. © 2013 Allerton Press, Inc
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A Note about Torsional Rigidity and Euclidean Moment of Inertia of Plane Domains
2020Co-Authors: Salakhudinov R.Abstract:© 2018, Pleiades Publishing, Ltd. Denote by P(G) the torsional rigidity of a Simply Connected plane Domain G, and by I2(G) the Euclidean moment of inertia of G. In 1995 F.G. Avkhadiev proved that P(G) and I2(G) are comparable quantities in sense of Pólya and Szegö. Moreover, it was shown that the ratio P(G) /I2(G) belongs to the segment [1, 64]. We investigate the following conjecture P(G) ≥ 3I2(G), where G is a Simply Connected Domain. We prove that the conjecture is true for polygonal Domains circumscribed about a circle. For convex Domains we show sharp isoperimetric inequalities, which justify the conjecture, in particular, we prove that P(G) > 2I2(G). Some aspects of approximate formulas for P(G) are also discussed
Filomena Pacella - One of the best experts on this subject based on the ideXlab platform.
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sign changing solutions of lane emden problems with interior nodal line and semilinear heat equations
Journal of Differential Equations, 2013Co-Authors: Francesca De Marchis, Isabella Ianni, Filomena PacellaAbstract:Abstract We consider the semilinear Lane Emden problem { − Δ u = | u | p − 1 u in Ω , u = 0 on ∂ Ω where Ω is a smooth bounded Simply Connected Domain in R 2 , invariant by the action of a finite symmetry group G. We show that if the orbit of each point in Ω, under the action of the group G, has cardinality greater than or equal to 4 then, for p sufficiently large, there exists a sign-changing solution of ( E p ) with two nodal regions whose nodal line does not touch ∂Ω. This result is proved as a consequence of an analogous result for the associated parabolic problem.
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sign changing solutions of lane emden problems with interior nodal line and semilinear heat equations
arXiv: Analysis of PDEs, 2012Co-Authors: Francesca De Marchis, Isabella Ianni, Filomena PacellaAbstract:We consider the semilinear Lane Emden problem in a smooth bounded Simply Connected Domain in the plane, invariant by the action of a finite symmetry group G. We show that if the orbit of each point in the Domain, under the action of the group G, has cardinality greater than or equal to four then, for p sufficiently large, there exists a sign changing solution of the problem with two nodal regions whose nodal line does not touch the boundary of the Domain. This result is proved as a consequence of an analogous result for the associated parabolic problem.
R G Salakhudinov - One of the best experts on this subject based on the ideXlab platform.
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isoperimetric properties of euclidean boundary moments of a Simply Connected Domain
Russian Mathematics, 2013Co-Authors: R G SalakhudinovAbstract:We consider integral functionals of a Simply Connected Domain which depend on the distance to the Domain boundary. We prove an isoperimetric inequality generalizing theorems derived by the Schwarz symmetrization method. For Lp-norms of the distance function we prove an analog of the Payne inequality for the torsional rigidity of the Domain. In compare with the Payne inequality we find new extremal Domains different from a disk.
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integral properties of the classical warping function of a Simply Connected Domain
Mathematical Notes, 2012Co-Authors: R G SalakhudinovAbstract:Let u(z,G) be the classical warping function of a Simply Connected Domain G. We prove that the Lp-norms of the warping function with different exponents are related by a sharp isoperimetric inequality, including the functional u(G) = supx∈Gu(x, G). A particular case of our result is the classical Payne inequality for the torsional rigidity of a Domain. On the basis of the warping function, we construct a new physical functional possessing the isoperimetric monotonicity property. For a class of integrals depending on the warping function, we also obtain a priori estimates in terms of the Lp-norms of the warping function as well as the functional u(G). In the proof, we use the estimation technique on level lines proposed by Payne.
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isoperimetric monotony of the l p norm of the warping function of a plane Simply Connected Domain
Russian Mathematics, 2010Co-Authors: R G SalakhudinovAbstract:Let G be a Simply Connected Domain and let u(x, G) be its warping function. We prove that L p -norms of functions u and u −1 are monotone with respect to the parameter p. This monotony also gives isoperimetric inequalities for norms that correspond to different values of the parameter p. The main result of this paper is a generalization of classical isoperimetric inequalities of St. Venant- P ´ olya and the Payne inequalities. DOI: 10.3103/S1066369X10080074
Hiroyuki Suzuki - One of the best experts on this subject based on the ideXlab platform.
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convergence of loop erased random walks on a planar graph to a chordal sle 2 curve
arXiv: Probability, 2014Co-Authors: Hiroyuki SuzukiAbstract:In this paper we consider the natural random walk on a planar graph and scale it by a small positive number $\delta$. Given a Simply Connected Domain $D$ and its two boundary points $a$ and $b$, we start the scaled walk at a vertex of the graph nearby $a$ and condition it on its exiting $D$ through a vertex nearby $b$, and prove that the loop erasure of the conditioned walk converges, as $\delta \downarrow 0$, to the chordal SLE$_{2}$ that connects $a$ and $b$ in $D$, provided that an invariance principle is valid for both the random walk and the dual walk of it.
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convergence of loop erased random walks on a planar graph to a chordal sle 2 curve
Kodai Mathematical Journal, 2014Co-Authors: Hiroyuki SuzukiAbstract:In this talk we consider the natural random walk on a planar graph and scale it by a small positive number δ. Given a Simply Connected Domain D and its two boundary points a and b, we start the scaled walk at a vertex of the graph nearby a and condition it on its exiting D through a vertex nearby b, and prove that the loop erasure of the conditioned walk converges, as δ ↓ 0, to the chordal SLE(2) curve that connects a and b in D, provided that an invariance principle is valid for both the random walk and the dual walk of it.
