Semilinear Dirichlet problem below the critical exponent #
Minimising the Dirichlet energy ∫ |∇u|² over the functions of unit L^q(Ω) norm on the unit
ball, for 2 ≤ q < 2⋆, produces a weak solution of
-Δu = λ|u|^{q-2}u, λ = ∫ |∇u|² > 0,
which is the equation of Guo's Section IX.1. EllipticPdes.Analysis.exists_bilin_minimiser
supplies the minimiser and EllipticPdes.Analysis.euler_lagrange_of_bilin_min supplies the
equation, with the Rellich compact embedding
EllipticPdes.Embedding.rellichEmbL_isCompact_of_lt as the only analytic input beyond
coercivity.
EllipticPdes.Embedding.exists_weakSolution_semilinear_of_lt runs the same two steps at the
graph norm of H₀¹(Ω) rather than at the Dirichlet energy, and reaches
-Δu + u = λ|u|^{q-2}u. Poincaré makes the two quadratics equivalent, so both minimisation
problems have solutions; their minimisers differ, and so do the equations. Guo states the
Dirichlet-energy one.
Main declarations #
EllipticPdes.Embedding.exists_dirichlet_minimiser_of_lt: the Dirichlet energy attains its minimum on the unitL^qsphere.EllipticPdes.Embedding.exists_weakSolution_dirichlet_of_lt: the minimiser solves-Δu = λ|u|^{q-2}u.EllipticPdes.Embedding.exists_weakSolution_dirichlet_of_lt': the same identity written out over the gradient coordinates.
References #
James Guo, Partial Differential Equations, Section IX.1; L. C. Evans, Partial Differential Equations (2nd ed.), §8.5.
Direct method at the Dirichlet energy. Below the critical exponent the Dirichlet
energy attains its minimum on the functions of unit L^q norm.
Semilinear Dirichlet problem. The minimiser of the Dirichlet energy on the unit
L^q sphere is a weak solution of -Δu = λ|u|^{q-2}u, with λ = ∫ |∇u|² the minimum itself,
which is positive.
Semilinear Dirichlet problem written over the gradient coordinates. The identity of
exists_weakSolution_dirichlet_of_lt with both sides unfolded: ∫ ∇u · ∇v = λ ∫ |u|^{q-2}uv for
every v ∈ H₀¹, with λ = ∫ |∇u|².