scholarly journals The p-Laplace Equation in Domains with Multiple Crack Section via Pencil Operators

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Pablo Álvarez-Caudevilla ◽  
Victor A. Galaktionov
Keyword(s):  
Laplace Equation ◽  
Blow Up ◽  
Nonlinear Eigenvalue Problem ◽  
Multiple Cracks ◽  
Harmonic Polynomials ◽  
Multiple Crack ◽  
Quasilinear Elliptic Problem ◽  
Nonlinear Eigenvalues ◽  
The One ◽  
Quasilinear Elliptic

AbstractThe p-Laplace equation∇ · (|∇u|in a bounded domain Ω ⊂ ℝΓ = Γmodeling a multiple crack formation, focusing at the origin 0 ∈ Ω. This makes the above quasilinear elliptic problem overdetermined. Possible types of the behaviour of solution u(x, y) at the tip 0 of such admissible multiple cracks, being a “singularity” point, are described, on the basis of blow-up scaling techniques and a “nonlinear eigenvalue problem” via spectral theory of pencils of non self-adjoint operators. Specially interesting is the application of those techniques to non-linear problems as the one considered here.To do so we introduce a very novel change of variable compared with the classical one introduced by Kondratiev for the analysis of non-smooth domains, such as domains with corner points, edges, etc, studying the behaviour of the solutions at those problematic points. Typical types of admissible cracks are shown to be governed by nodal sets of a countable family of nonlinear eigenfunctions, which are obtained via branching from harmonic polynomials that occur for p = 2. Using a combination of analytic and numerical methods, saddle-node bifurcations in p are shown to occur for those nonlinear eigenvalues/ eigenfunctions.

Quasilinear Equations With Boundary Blow-up and Exponential Reaction

2009 ◽  
Vol 9 (1) ◽  
Author(s):  
Jorge García-Melián
Keyword(s):  
Elliptic Problem ◽  
Blow Up ◽  
Quasilinear Equations ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic

AbstractWe consider the quasilinear elliptic problem Δ

Structure of boundary blow-up solutions for quasilinear elliptic problems I. Large and small solutions

2005 ◽  
Vol 135 (3) ◽  
pp. 615-642
Author(s):  
Zongming Guo ◽  
J. R. L. Webb
Keyword(s):  
Boundary Layer ◽  
Blow Up ◽  
Large Solution ◽  
Quasilinear Elliptic Problem ◽  
Boundary Layer Solution ◽  
Quasilinear Elliptic Problems ◽  
Bounded Smooth ◽  
Quasilinear Elliptic ◽  
Positive Boundary ◽  
Layer Solution

Existence and uniqueness of large, boundary blow-up solutions are obtained for the quasilinear elliptic problem −Δpu = λf(u) in Ω, u = ∞ on ∂Ω via good boundary layer estimates for large λ, where Δp is the p-Laplacian (1 < p < ∞) and Ω ⊂ ℝ N (N ≥ 2) is a bounded smooth domain. The nonlinear term f satisfies f(0) = f(z1) = f(z2) = 0 with 0 < z1 < z2, with z2 a zero of f of order k. It is shown that, if k ≥ p −1, the unique large solution ūλ is a boundary-layer solution which satisfies ūλ > z2 in Ω; if 0 < k < p −1, the unique large solution ūλ is a boundary-layer solution, but a flat core of ūλ occurs. Furthermore, for sufficiently large λ a small positive boundary blow-up solution uλ is obtained and its asymptotic behaviour as λ → ∞ is discussed.

Structure of boundary blow-up solutions for quasilinear elliptic problems I. Large and small solutions

2005 ◽  
Vol 135 (3) ◽  
pp. 615-642 ◽  
Author(s):  
Zongming Guo ◽  
J. R. L. Webb
Keyword(s):  
Boundary Layer ◽  
Blow Up ◽  
Large Solution ◽  
Quasilinear Elliptic Problem ◽  
Boundary Layer Solution ◽  
Quasilinear Elliptic Problems ◽  
Bounded Smooth ◽  
Quasilinear Elliptic ◽  
Positive Boundary ◽  
Layer Solution

Existence and uniqueness of large, boundary blow-up solutions are obtained for the quasilinear elliptic problem −Δpu = λf(u) in Ω, u = ∞ on ∂Ω via good boundary layer estimates for large λ, where Δp is the p-Laplacian (1 < p < ∞) and Ω ⊂ R N (N ≥ 2) is a bounded smooth domain. The nonlinear term f satisfies f(0) = f(z1) = f(z2) = 0 with 0 < z1 < z2, with z2 a zero of f of order k. It is shown that, if k ≥ p −1, the unique large solution ūλ is a boundary-layer solution which satisfies ūλ > z2 in Ω; if 0 < k < p −1, the unique large solution ūλ is a boundary-layer solution, but a flat core of ūλ occurs. Furthermore, for sufficiently large λ a small positive boundary blow-up solution is obtained and its asymptotic behaviour as λ → ∞ is discussed.

scholarly journals Gauge theories on compact toric manifolds

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Giulio Bonelli ◽  
Francesco Fucito ◽  
Jose Francisco Morales ◽  
Massimiliano Ronzani ◽  
Ekaterina Sysoeva ◽  
...  
Keyword(s):  
Partition Function ◽  
Gauge Theories ◽  
Blow Up ◽  
Gauge Coupling ◽  
Piecewise Constant Function ◽  
Partition Functions ◽  
Piecewise Constant ◽  
Toric Manifolds ◽  
Holomorphic Anomaly ◽  
The One

AbstractWe compute the $$\mathcal{N}=2$$ N = 2 supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kähler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on $$\mathbb {C}^2$$ C 2 . The evaluation of these residues is greatly simplified by using an “abstruse duality” that relates the residues at the poles of the one-loop and instanton parts of the $$\mathbb {C}^2$$ C 2 partition function. As particular cases, our formulae compute the SU(2) and SU(3) equivariant Donaldson invariants of $$\mathbb {P}^2$$ P 2 and $$\mathbb {F}_n$$ F n and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the SU(2) case. Finally, we show that the U(1) self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a $$\mathcal {N}=2$$ N = 2 analog of the $$\mathcal {N}=4$$ N = 4 holomorphic anomaly equations.

MULTIPLE POSITIVE SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS ASSOCIATED TO A CLASS OF p-LAPLACIAN LIKE OPERATORS

2003 ◽  
Vol 05 (05) ◽  
pp. 737-759 ◽  
Author(s):  
NOBUYOSHI FUKAGAI ◽  
KIMIAKI NARUKAWA
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Eigenvalue Problems ◽  
Multiple Positive Solutions ◽  
Combined Effects ◽  
Nonlinear Eigenvalue Problems ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic ◽  
Nonlinear Eigenvalue

This paper deals with positive solutions of a class of nonlinear eigenvalue problems. For a quasilinear elliptic problem (#) - div (ϕ(|∇u|)∇u) = λf(x,u) in Ω, u = 0 on ∂Ω, we assume asymptotic conditions on ϕ and f such as ϕ(t) ~ tp0-2, f(x,t) ~ tq0-1as t → +0 and ϕ(t) ~ tp1-2, f(x,t) ~ tq1-1as t → ∞. The combined effects of sub-nonlinearity (p0> q0) and super-nonlinearity (p1< q1) with the subcritical term f(x,u) imply the existence of at least two positive solutions of (#) for 0 < λ < Λ.

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