| 1. |
- Aghajani, Asadollah, et al.
(författare)
-
Pointwise estimates for systems of coupled p-laplacian elliptic equations
- 2023
-
Ingår i: Communications on Pure and Applied Analysis. - : American Institute of Mathematical Sciences. - 1534-0392 .- 1553-5258. ; 22:3, s. 899-921
-
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.
|
|
| 2. |
|
|
| 3. |
- Arakelyan, A., et al.
(författare)
-
Two-and multi-phase quadrature surfaces
- 2017
-
Ingår i: Communications on Pure and Applied Analysis. - : American Institute of Mathematical Sciences. - 1534-0392 .- 1553-5258. ; 16:6, s. 2023-2045
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we shall initiate the study of the two-and multi-phase quadrature surfaces (QS), which amounts to a two/multi-phase free boundary problems of Bernoulli type. The problem is studied mostly from a potential theoretic point of view that (for two-phase case) relates to integral representation where dsx is the surface measure, μ = μ+-μ-is given measure with support in (a priori unknown domain) ω = ω+ [ω-, g is a given smooth positive function, and the integral holds for all functions h, which are harmonic on ω. Our approach is based on minimization of the corresponding two-and multiphase functional and the use of its one-phase version as a barrier. We prove several results concerning existence, qualitative behavior, and regularity theory for solutions. A central result in our study states that three or more junction points do not appear.
|
|
| 4. |
- Baravdish, George, 1964-, et al.
(författare)
-
Generalizations of p-Laplace operator for image enhancement : Part 2
- 2020
-
Ingår i: Communications on Pure and Applied Analysis. - Springfield, MO, United States : American Institute of Mathematical Sciences. - 1534-0392 .- 1553-5258. ; 19:7, s. 3477-3500
-
Tidskriftsartikel (refereegranskat)abstract
- We have in a previous study introduced a novel elliptic operator Delta((p,q))u = vertical bar del u vertical bar(q) Delta(1)u + (p - 1) vertical bar del u vertical bar(p-2)Delta(infinity) u, p >= 1, q >= 0, as a generalization of the p-Laplace operator. In this paper, we establish the well-posedness of the parabolic equation u(t) =vertical bar del u vertical bar(1-q) Delta((1+q,q)), where q = q(vertical bar del u vertical bar) is continuous and has range in [0, 1], in the framework of viscosity solutions. We prove the consistency and convergence of the numerical scheme of finite differences of this parabolic equation. Numerical simulations shows the advantage of this operator applied to image enhancement.
|
|
| 5. |
- Cianchi, Andrea, et al.
(författare)
-
GLOBAL GRADIENT ESTIMATES IN ELLIPTIC PROBLEMS UNDER MINIMAL DATA AND DOMAIN REGULARITY
- 2015
-
Ingår i: Communications on Pure and Applied Analysis. - : American Institute of Mathematical Sciences (AIMS). - 1534-0392 .- 1553-5258. ; 14:1, s. 285-311
-
Tidskriftsartikel (refereegranskat)abstract
- This is a survey of some recent contributions by the authors on global integrability properties of the gradient of solutions to boundary value problems for nonlinear elliptic equations in divergence form. Minimal assumptions on the regularity of the ground domain and of the prescribed data are pursued.
|
|
| 6. |
- Euler, Marianna, et al.
(författare)
-
Integrating factors and conservation laws for some Camassa-Holm type equations
- 2012
-
Ingår i: Communications on Pure and Applied Analysis. - : American Institute of Mathematical Sciences (AIMS). - 1534-0392 .- 1553-5258. ; 11:4, s. 1421-1430
-
Tidskriftsartikel (refereegranskat)abstract
- We classify all 1st-order integrating factors and the correspondingconservation laws for a class of Camassa-Holm type equations.
|
|
| 7. |
- Kennedy, James B., et al.
(författare)
-
On the hot spots of quantum graphs
- 2021
-
Ingår i: Communications on Pure and Applied Analysis. - : American Institute of Mathematical Sciences (AIMS). - 1534-0392 .- 1553-5258. ; 20:9, s. 3029-3063
-
Tidskriftsartikel (refereegranskat)abstract
- We undertake a systematic investigation of the maxima and minima of the eigenfunctions associated with the first nontrivial eigenvalue of the Laplacian on a metric graph equipped with standard (continuity–Kirchhoff) vertex conditions. This is inspired by the famous hot spots conjecture for the Laplacian on a Euclidean domain, and the points on the graph where maxima and minima are achieved represent the generically "hottest" and "coldest" spots of the graph. We prove results on both the number and location of the hot spots of a metric graph, and also present a large number of examples, many of which run contrary to what one might naïvely expect. Amongst other results we prove the following: (i) generically, up to arbitrarily small perturbations of the graph, the points where minimum and maximum, respectively, are attained are unique; (ii) the minima and maxima can only be located at the vertices of degree one or inside the doubly connected part of the metric graph; and (iii) for any fixed graph topology, for some choices of edge lengths all minima and maxima will occur only at degree-one vertices, while for others they will only occur in the doubly connected part of the graph.
|
|
| 8. |
- Mikayelyan, Hayk, et al.
(författare)
-
Convexity Of The Free Boundary For An Exterior Free Boundary Problem Involving The Perimeter
- 2013
-
Ingår i: Communications on Pure and Applied Analysis. - : American Institute of Mathematical Sciences (AIMS). - 1534-0392 .- 1553-5258. ; 12:3, s. 1431-1443
-
Tidskriftsartikel (refereegranskat)abstract
- We prove that if the given compact set K is convex then a minimizer of the functional I(v) = integral(BR) vertical bar del v vertical bar(p)dx + Per({v > 0}), 1 < p < infinity, over the set {v epsilon W-0(1,P)(BR)vertical bar v 1 on K subset of BR} has a convex support, and as a result all its level sets are convex as well. We derive the free boundary condition for the minimizers and prove that the free boundary is analytic and the minimizer is unique.
|
|
| 9. |
- Strömberg, Thomas
(författare)
-
A system of the Hamilton-Jacobi and the continuity equations in the vanishing viscosity limit
- 2011
-
Ingår i: Communications on Pure and Applied Analysis. - : American Institute of Mathematical Sciences (AIMS). - 1534-0392 .- 1553-5258. ; 10:2, s. 479-506
-
Tidskriftsartikel (refereegranskat)abstract
- We study the following system of the viscous Hamilton-Jacobi and the continuity equations in the limit as epsilon down arrow 0: S-t(epsilon) + 1/2 vertical bar DS epsilon vertical bar(2) + V(x) - epsilon Delta S-epsilon = 0 in Q(T), S-epsilon(0, x) = S-0(x) in R-n; rho(epsilon)(t) + div(rho(epsilon) DS epsilon) = 0 in Q(T), rho(epsilon)(0, x) = rho(0)(x) in R-n. Here Q(T) = (0, T] x R-n. The potential V and the initial function S-0 are allowed to grow quadratically while rho(0) is a Borel measure. The paper justifies and describes the vanishing viscosity transition to the corresponding inviscid system. The notion of weak solution employed for the inviscid system is that of a viscosity-measure solution (S, rho).
|
|
