On the Equivalence of Viscosity Solutions and Weak Solutions for a Quasi-Linear Equation

Juutinen, Petri; Lindqvist, Peter; Manfredi, Juan J.

doi:10.1137/s0036141000372179

Cited by 231 publications

(222 citation statements)

References 15 publications

(2 reference statements)

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“…This definition coincides with the definition given earlier when 2 ≤ p < ∞. (See also [13] and [4] for a further discussion.) (1) u is lower semicontinuous.…”

Section: P-superharmonic Functionssupporting

confidence: 66%

A sub-Riemannian maximum principle and its application to the p-Laplacian in Carnot groups

Bieske¹

2012

Ann. Acad. Sci. Fenn. Math.

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Abstract. We prove a sub-Riemannian maximum principle for semicontinuous functions. We apply this principle to Carnot groups to provide a "sub-Riemannian" proof of the uniqueness of viscosity infinite harmonic functions. This is an alternate method of proof from the one found in [15]. We also establish the equivalence of weak solutions and viscosity solutions to the p-Laplace equation. This result extends the author's previous work in the Heisenberg group [3, 4]. Calculus on Carnot groupsWe begin by denoting an arbitrary Carnot group in R N by G and its corresponding Lie Algebra by g. Recall that g is nilpotent and stratified, resulting in the decompositionfor appropriate vector spaces that satisfy the Lie bracket relation [V 1 , V j ] = V 1+j . The Lie Algebra g is associated with the group G via the exponential map exp : g → G.Since this map is a diffeomorphism, we can choose a basis for g so that it is the identity map. Denote this basis byWe endow g with an inner product ·, · and related norm · so that this basis is orthonormal. Clearly, the Riemannian dimension of g (and so G) is N = n 1 +n 2 +n 3 . However, we will also consider the homogeneous dimension of G, denoted Q, which is given by

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“…This definition coincides with the definition given earlier when 2 ≤ p < ∞. (See also [13] and [4] for a further discussion.) (1) u is lower semicontinuous.…”

Section: P-superharmonic Functionssupporting

confidence: 66%

A sub-Riemannian maximum principle and its application to the p-Laplacian in Carnot groups

Bieske¹

2012

Ann. Acad. Sci. Fenn. Math.

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“…It is also worth noting that superparabolic functions to the evolutionary p-Laplace equation coincide with the viscosity supersolutions, as Juutinen, Lindqvist and Manfredi proved in [8].…”

Section: Introductionmentioning

confidence: 78%

A connection between a general class of superparabolic functions and supersolutions

2009

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“…This idea, which dates back to Aronsson, has been effectively put into action by applying the theory of viscosity solutions to the ∞-Laplacian (see e.g. [12,19] and references therein) which, in view of the equivalence of weak and viscosity solutions [18,31] for the p finite case, is very stable under limits. Further, in view of the uniqueness in the scalar case, all subsequential limits as p → ∞ give rise to a viscosity solution of the limit equation.…”

Section: P Approximations As P → ∞ Of the L ∞ Equationsmentioning

confidence: 99%

On the numerical approximation of $$\infty $$ ∞ -harmonic mappings

Katzourakis

Pryer

2016

Nonlinear Differ. Equ. Appl.

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Abstract. A map u : Ω ⊆ R n −→ R N , is said to be ∞-harmonic if it satisfies(1)The system (1) is the model of vector-valued Calculus of Variations in L ∞ and arises as the "Euler-Lagrange" equation in relation to the supremal functional(2)In this work we provide numerical approximations of solutions to the Dirichlet problem when n = 2 and in the vector valued case of N = 2, 3 for certain carefully selected boundary data on the unit square. Our experiments demonstrate interesting and unexpected phenomena occurring in the vector valued case and provide insights on the structure of general solutions and the natural separation to phases they present.

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On the Equivalence of Viscosity Solutions and Weak Solutions for a Quasi-Linear Equation

Cited by 231 publications

References 15 publications

A sub-Riemannian maximum principle and its application to the p-Laplacian in Carnot groups

A sub-Riemannian maximum principle and its application to the p-Laplacian in Carnot groups

A connection between a general class of superparabolic functions and supersolutions

On the numerical approximation of $$\infty $$ ∞ -harmonic mappings

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