| 1. |
- Acker, Andrew, et al.
(författare)
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Convex configurations for solutions to semilinear elliptic problems in convex rings
- 2006
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Ingår i: Communications in Partial Differential Equations. - : Informa UK Limited. - 0360-5302 .- 1532-4133. ; 31:9, s. 1273-1287
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Tidskriftsartikel (refereegranskat)abstract
- For a given convex ring Omega = Omega(2)\(Omega) over bar (1) and an L-1 function f : Omega x R -> R+ we show, under suitable assumptions on f, that there exists a solution (in the weak sense) to Delta(p)u = f(x, u) in Omega u = 0 on partial derivative Omega(2) u = M on partial derivative Omega(1) with {x is an element of Omega : u(x) > s} boolean OR Omega(1) convex, for all s is an element of (0, M).
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| 2. |
- Acker, A., et al.
(författare)
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The multi-layer free boundary problem for the p-Laplacian in convex domains
- 2004
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Ingår i: Interfaces and free boundaries (Print). - 1463-9963 .- 1463-9971. ; 6:1, s. 81-103
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Tidskriftsartikel (refereegranskat)abstract
- The main result of this paper concerns existence of classical solutions to the multi-layer Bernoulli free boundary problem with nonlinear joining conditions and the p-Laplacian as governing operator. The present treatment of the two-layer case involves technical refinements of the one-layer case, studied earlier by two of the authors. The existence treatment of the multi-layer case is largely based on a reduction to the two-layer case, in which uniform separation of the free boundaries plays a key role.
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| 3. |
- Aghajani, Asadollah, et al.
(författare)
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Pointwise estimates for systems of coupled p-laplacian elliptic equations
- 2023
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Ingår i: Communications on Pure and Applied Analysis. - : American Institute of Mathematical Sciences. - 1534-0392 .- 1553-5258. ; 22:3, s. 899-921
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Tidskriftsartikel (refereegranskat)abstract
- This work examines positive solutions of systems of inequalities ±∆pu ≥ ρ(x)f (u), in Ω, where p = (p1, ..., pk), pi > 1 and ∆p is the diagonal-matrix diag(∆p1 , ..., ∆pk ), ∆pi is the pi-Laplace operator, Ω is an arbitrary domain (bounded or not) in RN (N ≥ 2), u = (u1, ..., uk)T and f = (f1, ..., fk)T are vector-valued functions and ρ(x) is a nonnegative function in Ω which is locally bounded. Using a maximum principle-based argument we provide explicit estimates on positive solutions u at each point x ∈ Ω, and as applications we find Liouville type results in unbounded domains such as RN, exterior domains or generally unbounded domains with the property that supx∈Ω dist(x, ∂Ω) = ∞, for various nonlinearities f and weights ρ. We also give explicit upper bounds on extremal parameters of related nonlinear multi-parameter eigenvalue problems in bounded domains.
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| 4. |
- Aleksanyan, G., et al.
(författare)
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Regularity of the free boundary for a parabolic cooperative system
- 2022
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Ingår i: Calculus of Variations and Partial Differential Equations. - : Springer Nature. - 0944-2669 .- 1432-0835. ; 61:4
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study the following parabolic system Delta u - partial derivative(t)u = vertical bar u vertical bar(q-1) u chi({vertical bar u vertical bar > 0}), = (u(1), ... , u(m)), with free boundary partial derivative{vertical bar u vertical bar > 0). For 0 <= q < 1, we prove optimal growth rate for solutions u to the above system near free boundary points, and show that in a uniform neighbourhood of any a priori well-behaved free boundary point the free boundary is C-1,C-alpha in space directions and half-Lipschitz in the time direction.
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| 5. |
- Aleksanyan, Hayk, et al.
(författare)
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Applications of Fourier analysis in homogenization of Dirichlet problem I. Pointwise estimates
- 2013
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Ingår i: Journal of Differential Equations. - : Elsevier BV. - 0022-0396 .- 1090-2732. ; 254:6, s. 2626-2637
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex and smooth domain, and smooth operator and boundary data, we prove pointwise convergence results, namely vertical bar u(epsilon)(x) - u(0)(x)vertical bar <= C-kappa epsilon((d-1)/2) 1/d(x)(kappa), for all x is an element of D, for all kappa > d - 1, where u(epsilon) and u(0) are solutions of respectively oscillating and homogenized Dirichlet problems, and d(x) is the distance of x from the boundary of D. As a corollary for all 1 <= p < infinity we obtain L-P convergence rate as well.
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| 6. |
- Aleksanyan, Hayk, et al.
(författare)
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Applications of Fourier Analysis in Homogenization of Dirichlet Problem III : Polygonal Domains
- 2014
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Ingår i: Journal of Fourier Analysis and Applications. - : Springer. - 1069-5869 .- 1531-5851. ; 20:3, s. 524-546
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we prove convergence results for the homogenization of the Dirichlet problem for elliptic equations in divergence form with rapidly oscillating boundary data and non oscillating coefficients in convex polygonal domains. Our analysis is based on integral representation of solutions. Under a certain Diophantine condition on the boundary of the domain and smooth coefficients we prove pointwise, as well as convergence results. For larger exponents we prove that the convergence rate is close to optimal. We also suggest several directions of possible generalization of the results in this paper.
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| 7. |
- Aleksanyan, Hayk, et al.
(författare)
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Applications of Fourier Analysis in Homogenization of the Dirichlet Problem : L-p Estimates
- 2015
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Ingår i: Archive for Rational Mechanics and Analysis. - : Springer. - 0003-9527 .- 1432-0673. ; 215:1, s. 65-87
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Tidskriftsartikel (refereegranskat)abstract
- Let u(epsilon) be a solution to the system div(A(epsilon)(x)del u(epsilon)(x)) = 0 in D, u(epsilon)(x) = g(x, x/epsilon) on partial derivative D, where D subset of R-d (d >= 2), is a smooth uniformly convex domain, and g is 1-periodic in its second variable, and both A(epsilon) and g are sufficiently smooth. Our results in this paper are twofold. First we prove L-p convergence results for solutions of the above system and for the non-oscillating operator A(epsilon)(x) = A(x), with the following convergence rate for all 1 <= p < infinity parallel to u(epsilon) - u(0)parallel to (LP(D)) <= C-P {epsilon(1/2p), d = 2, (epsilon vertical bar ln epsilon vertical bar)(1/p), d = 3, epsilon(1/p), d >= 4, which we prove is (generically) sharp for d >= 4. Here u(0) is the solution to the averaging problem. Second, combining our method with the recent results due to Kenig, Lin and Shen (Commun Pure Appl Math 67(8): 1219-1262, 2014), we prove (for certain class of operators and when d >= 3) ||u(epsilon) - u(0)||(Lp(D)) <= C-p[epsilon(ln(1/epsilon))(2)](1/p) for both the oscillating operator and boundary data. For this case, we take A(epsilon) = A(x/epsilon), where A is 1-periodic as well. Some further applications of the method to the homogenization of the Neumann problem with oscillating boundary data are also considered.
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| 8. |
- Aleksanyan, Hayk, et al.
(författare)
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Discrete balayage and boundary sandpile
- 2019
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Ingår i: Journal d'Analyse Mathematique. - : Springer. - 0021-7670 .- 1565-8538. ; 138:1, s. 361-403
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a new lattice growth model, which we call the boundary sandpile. The model amounts to potential-theoretic redistribution of a given initial mass on Z(d) (d >= 2) onto the boundary of an (a priori) unknown domain. The latter evolves through sandpile dynamics, and has the property that the mass on the boundary is forced to stay below a prescribed threshold. Since finding the domain is part of the problem, the redistribution process is a discrete model of a free boundary problem, whose continuum limit is yet to be understood. We prove general results concerning our model. These include canonical representation of the model in terms of the smallest super-solution among a certain class of functions, uniform Lipschitz regularity of the scaled odometer function, and hence the convergence of a subsequence of the odometer and the visited sites, discrete symmetry properties, as well as directional monotonicity of the odometer function. The latter (in part) implies the Lipschitz regularity of the free boundary of the sandpile.As a direct application of some of the methods developed in this paper, combined with earlier results on the classical abelian sandpile, we show that the boundary of the scaling limit of an abelian sandpile is locally a Lipschitz graph.
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| 9. |
- Aleksanyan, Hayk, et al.
(författare)
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L2-estimates for singular oscillatory integral operators
- 2016
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Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier. - 0022-247X .- 1096-0813. ; 441:2, s. 529-548
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Tidskriftsartikel (refereegranskat)abstract
- In this note we study singular oscillatory integrals with linear phase function over hypersurfaces which may oscillate, and prove estimates of L2L2 type for the operator, as well as for the corresponding maximal function. If the hypersurface is flat, we consider a particular class of a nonlinear phase functions, and apply our analysis to the eigenvalue problem associated with the Helmholtz equation in R3.
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| 10. |
- Aleksanyan, Hayk, et al.
(författare)
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Perturbed Divisible Sandpiles and Quadrature Surfaces
- 2019
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Ingår i: Potential Analysis. - : SPRINGER. - 0926-2601 .- 1572-929X. ; 51:4, s. 511-540
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Tidskriftsartikel (refereegranskat)abstract
- The main purpose of the present paper is to establish a link between quadrature surfaces (potential theoretic concept) and sandpile dynamics (Laplacian growth models). For this aim, we introduce a new model of Laplacian growth on the lattice DOUBLE-STRUCK CAPITAL Zd (d >= 2) which continuously deforms occupied regions of the divisible sandpile model of Levine and Peres (J. Anal. Math. 111(1), 151-219 2010), by redistributing the total mass of the system onto 1/m-sub-level sets of the odometer which is a function counting total emissions of mass from lattice vertices. In free boundary terminology this goes in parallel with singular perturbation, which is known to converge to a Bernoulli type free boundary. We prove that models, generated from a single source, have a scaling limit, if the threshold m is fixed. Moreover, this limit is a ball, and the entire mass of the system is being redistributed onto an annular ring of thickness 1/m. By compactness argument we show that when m tends to infinity sufficiently slowly with respect to the scale of the model, then in this case also there is scaling limit which is a ball, with the mass of the system being uniformly distributed onto the boundary of that ball, and hence we recover a quadrature surface in this case. Depending on the speed of decay of 1/m, the visited set of the sandpile interpolates between spherical and polygonal shapes. Finding a precise characterisation of this shape-transition phenomenon seems to be a considerable challenge, which we cannot address at this moment.
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