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

By Petri Juutinen, Peter Lindqvist, Juan J. Manfredi. SIAM Journal on Mathematical Analysis, 2001

318 citations recorded by the source corpus.

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In Applied Mathematics

Citations in the corpus

First eigenvalue and maximum principle for fully nonlinear singular operators (2004, background): “In fact in [ 18 , 19] Juutinen, Lindqvist and Manfredi opened the way to this topic.”
Harnack's Inequality for p-Harmonic Functions via Stochastic Games (2012, background): “Moreover, viscosity solutions to p-Laplacian coincide with the classical ones by [9] (see [8] for easier proof of this fact).”
A class of integral equations and approximation of p-Laplace equations (2010, result): “We refer to [ 12 ]f or results and proofs similar to those in this section.”
A connection between a general class of superparabolic functions and supersolutions (2010, result): “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].”
Non-local gradient dependent operators (2012, unknown): “We recall that the viscosity definition of ∆p,+ coincides with the variational one, see [11].”
Solutions of nonlinear PDEs in the sense of averages (2012, unknown): “According to [12], when 1 < p < ∞ and g ≡ 0, it is enough to test using test functions with ∇φ(x) ̸= 0.”
Harnack inequality for singular fully nonlinear operators and some existence results (2010, background): “We refer the reader to [28] for a related work on singular operators.”
Quasilinear and Hessian equations of Lane-Emden type (2005, background): “renormalized (entropy) sense for quasilinear equations, and viscosity, or k-convexity sense, for fully nonlinear equations of Hessian type (see [BMMP], [DMOP], [ JLM ], [TW1]‐[TW3], [Ur]).”
Alexandroff–Bakelman–Pucci estimate for singular or degenerate fully nonlinear elliptic equations (2009, methodology): “Notice that some similar notion of viscosity solution are used in [10], [13] and [15].”
A New Proof for the Equivalence of Weak and Viscosity Solutions for the p-Laplace Equation (2011, background): “In [8], the converse was established by showing, roughly speaking, that the Dirichlet boundary value problem has a unique viscosity solution.”
A Visit with the ∞-Laplace Equation (2008, background): “See Juutinen, Lindqvist and Manfredi, [39] and Ishii, [35].”
Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation (2006, background): “The p-superparabolic functions are of actual interest also because they are the viscosity supersolutions of (1.1), see [ JLM ].”