Uniqueness and continuum of foliated solutions for a quasilinear elliptic equation with a non lipschitz nonlinearity

Barles, Guy; Alcolea-Díaz, Gema; Díaz, J. I.

doi:10.1080/03605309208820876

Cited by 24 publications

(13 citation statements)

References 3 publications

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“…(for p ¼ 2; this example is due to Barles et al [3]). It is of further interest in connection with this example that the compact support principle can fail even if (1.8) is satisfied, namely when q4p À 1!…”

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confidence: 96%

The strong maximum principle revisited

Pucci

Serrin

2004

Journal of Differential Equations

239

211

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In this paper we first present the classical maximum principle due to E. Hopf, together with an extended commentary and discussion of Hopf's paper. We emphasize the comparison technique invented by Hopf to prove this principle, which has since become a main mathematical tool for the study of second order elliptic partial differential equations and has generated an enormous number of important applications. While Hopf's principle is generally understood to apply to linear equations, it is in fact also crucial in nonlinear theories, such as those under consideration here.In particular, we shall treat and discuss recent generalizations of the strong maximum principle, and also the compact support principle, for the case of singular quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. Our principal interest is in necessary and sufficient conditions for the validity of both principles; in exposing and simplifying earlier proofs of corresponding results; and in extending the conclusions to wider classes of singular operators than previously considered.The results have unexpected ramifications for other problems, as will develop from the exposition, e.g.(i) two point boundary value problems for singular quasilinear ordinary differential equations (Sections 3 and 4); (ii) the exterior Dirichlet boundary value problem (Section 5); (iii) the existence of dead cores and compact support solutions, i.e. dead cores at infinity (Section 7); (iv) Euler-Lagrange inequalities on a Riemannian manifold (Section 9); (v) comparison and uniqueness theorems for solutions of singular quasilinear differential inequalities (Section 10).

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confidence: 96%

The strong maximum principle revisited

Pucci

Serrin

2004

Journal of Differential Equations

239

211

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“…The first order equation cxjuj kÀ1 u axjDuj m À f x 0 26 is related to the eikonal equation and to deterministic optimal control theory, see [1], where c^0Y k 1 and aY f b 0X The SMaxP is false because, for instance, Àjxj is a viscosity solution of jDuj À 1 0Y in R N X The nondegeneracy condition (4) for the SMinP is satisfied for The condition m^1 in (28) is sharp if b À1 by an example in [17]. Uniqueness and nonuniqueness of solutions of the Dirichlet problem for the equation (27) with nonzero right-hand side and c 0Y b 1Y are studied in [3]. E x a m p l e 7 .…”

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confidence: 99%

On the strong maximum principle for fully nonlinear degenerate elliptic equations

Bardi

Lio

1999

Archiv der Mathematik

107

137

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We prove a Strong Maximum Principle for semicontinuous viscosity subsolutions or supersolutions of fully nonlinear degenerate elliptic PDEs, which complements the results of [17]. Our assumptions and conclusions are different from those in [17], in particular our maximum principle implies the nonexistence of a dead core. We test the assumptions on several examples involving the p-Laplacian and the minimal surface operator, and they turn out to be sharp in all cases where the existence of a dead core is known. We can also cover equations that are singular for p 0 and very degenerate operators such as the I -Laplacian and some first order Hamilton-Jacobi operators.

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“…The competition between the diffusion and the source term is already revealed by the structure of steady states to (1.1). Indeed, while it follows from Theorem 1 in [4], that zero is the only steady state in C(B) when p 2 and q p − 1, several steady states may exist when p 2 and q ∈ (0, p − 1) [6,14,20]. Another typical feature of the competition between diffusion and source is the possibility of finite time blow-up in a suitable norm, and this phenomenon has been shown to occur for (1.1) when p = 2 and q > 2, see [16] and the references therein.…”

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Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton–Jacobi equation

Barles

Laurençot

Stinner

2010

Asymptotic Analysis

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Convergence to a single steady state is shown for non-negative and radially symmetric solutions to a diffusive Hamilton-Jacobi equation with homogeneous Dirichlet boundary conditions, the diffusion being the p-Laplacian operator, p 2, and the source term a power of the norm of the gradient of u. As a first step, the radially symmetric and non-increasing stationary solutions are characterized. G. Barles et al. / Convergence to steady states for radially symmetric solutionsWe further assume the initial condition u 0 ∈ W 1,∞ 0 (B) is radially symmetric and non-negative and u 0 ≡ 0, (1.2) while the parameters p and q satisfy p 2 and 0 < q < p − 1.(1.3)The partial differential equation in (1.1) is a second-order parabolic equation featuring a diffusion term (possibly quasilinear and degenerate if p > 2) and a source term |∇u| q counteracting the effect of diffusion and depending solely on the gradient of the solution. The competition between the diffusion and the source term is already revealed by the structure of steady states to (1.1). Indeed, while it follows from Theorem 1 in [4], that zero is the only steady state in C(B) when p 2 and q p − 1, several steady states may exist when p 2 and q ∈ (0, p − 1) [6,14,20]. Another typical feature of the competition between diffusion and source is the possibility of finite time blow-up in a suitable norm, and this phenomenon has been shown to occur for (1.1) when p = 2 and q > 2, see [16] and the references therein. More precisely, it is established in [18] that, when p = 2 and q > 2, there are classical solutions to (1.1) for which the L ∞ -norm of the gradient blows up in finite time, the L ∞ -norm of the solution remaining bounded. These solutions may actually be extended to all positive times in a unique way within the framework of viscosity solutions [5,21], the boundary condition being also satisfied in the viscosity sense. According to the latter, the homogeneous Dirichlet boundary condition might not always be fulfilled for all times, a property which is likely to be connected with the finite time blow-up of the gradient.Coming back to the case where p and q fulfil (1.3) and several steady states may exist, a complete classification of steady states seems to be out of reach when B is replaced by an arbitrary open set of R N . Nevertheless, there are at least two situations in which the set of stationary solutions can be described, namely, when N = 1 and B = (−1, 1) [14,20] and when N 2 under the additional requirement that the steady states are radially symmetric and non-increasing, the latter being the first result of this paper. More precisely, we show that (1.1) has a one-parameter family of stationary solutions and that each stationary solution is characterized by the value of its maximum. Theorem 1.1. Assume (1.3). Let w ∈ W 1,∞ (B) be a radially symmetric and non-increasing viscosity solution to −Δ p w − |∇w| q = 0 in B satisfying w = 0 on ∂B. Then there is ϑ ∈ [0, 1] such that w = w ϑ , where w ϑ (x) := c 0

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Uniqueness and continuum of foliated solutions for a quasilinear elliptic equation with a non lipschitz nonlinearity

Cited by 24 publications

References 3 publications

The strong maximum principle revisited

The strong maximum principle revisited

On the strong maximum principle for fully nonlinear degenerate elliptic equations

Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton–Jacobi equation

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