Sobolev spaces defined on non-compact Lie groups

In this post, a question was raised to discuss the generalization of Sobolev spaces on locally compact Lie groups. Now my question is whether there exists a generalization of Sobolev spaces and associated theories of the Laplace operator on noncompact Lie groups. My expectation is that the theory should be reduced to coincide with that … Read more

Approximation of monotone Sobolev functions

Let f∈W1,2loc(R2) be a continuous monotone (real valued) function (monotone in the sense that the maximum and minimum of f in a precompact open set are attained at the boundary). Is it true that there exists a sequence of smooth monotone functions fn converging to f in W1,2loc(R2)? We could ask the same question for … Read more

How can I can derive an explicit bound for the solution of the poisson’s PDE?

i need some help on this question Let Ω be an open subset of R2 (say a square) with ∂Ω=Γ1∪Γ2∪Γ3∪Γ4. A structure ocupying this surface is subject to : 1) A weight force F, i am given the explicit expression at each point of Ω . 2) Pressure force on Γ1, i am given the … Read more

Gelfand triple (H1+s,H1,H1−s)(H^{1+s}, H^1, H^{1-s})

I think I found a proof, that on Lipschitz domains Ω, H1+s(Ω) is the dual space of H1−s(Ω) with respect to the H1(Ω) scalar product for all 0≤s<1/2. Does anyone know a reference to such a result? (I am assuming that Hr(Ω) is defined by interpolation between L2(Ω), H1(Ω), and H2(Ω)). Answer AttributionSource : Link … Read more

inverse of sobolev riemannian metric still sobolev?

Given a covariant riemannian metric of certain sobolev class (i.e. with square-integrable weak derivatives up to a large enough integer order) on a (compact, if necessary) finite-dimensional smooth manifold, I would like to know the answers to the following related questions: (1) does its inverse (i.e. the contra-variant type) belong to the same sobolev class? … Read more

A reference for ∇|u|p=p sgn(u)|u|p−1∇u\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u

Let Ω be an open domain with nice boundary and u∈W1,p(Ω). I believe that |u|p∈W1,1 with ∇|u|p=p sgn(u)|u|p−1∇u but couldn’t find a good reference for this except for the case p=1,2. The assumption in standard textbooks that I usually see regarding the chain rule for Sobolev functions, i.e. ∇F(u)=F′(u)∇u, is that F∈C1 with bounded derivative (or … Read more

Can we approximate harmonic maps which are a.e. orientation-preserving with maps which preserve orientation globally?

Let Dn be the closed unit ball, and let f:Dn→Rn be harmonic; More precisely, I assume that f is real-analytic and harmonic on the interior (Dn)o and smooth on the entire closed ball. Suppose also that n≥2, and that detdf>0 a.e. on Dn. Are there harmonic maps ωn:Dn→Rn, such that detdωn>0 everywhere on Dn, and … Read more

Fractional Hajłasz-Besov-like similar to the Korevaar-Schoen-Sobolev spaces?

Suppose that (X,μ,d) and (Y,ν,ρ) are doubling metric measure spaces. Fix α>0 and define the space, analogously to this paper, as the collection of all measurable functions f:X→Y satisfying: (∫∞0[∫y∈Q∫x∈Qρ(f(x),f(y))pμ(B(x,t))αdμ(x)dμ(y)]pq1t1+sqdt)q<∞ Then the functions satisfying the above constraint can be seen as a non-Euclidean analogue of Hajłasz-Besov spaces, similar to the Korevaar-Shoen extensions of the Sobolev … Read more

Is Δϕ\Delta \phi monotone operator on H1(Rd)H^1(\mathbb{R}^d) for monotone ϕ\phi

Let H1(Rd) be the usual Sobolev space and let ϕ:R→R be a non decreasing Lipschitz function with ϕ(0)=0. Is the operator Δϕ on H1(Rd) monotone? i.e. Do we have ⟨Δϕ(u)−Δϕ(v),u−v⟩≥0( or ≤) for u,v∈H1(Rd)? This is easily verified for ϕ(x)=x, but I’m not able to conclude anything for general ϕ. ⟨Δϕ(u)−Δϕ(v),u−v⟩=−⟨∇ϕ(u)−∇ϕ(v),∇u−∇v⟩=−⟨ϕ′(u)∇u−ϕ′(v)∇v,∇u−∇v⟩ If ϕ′ is constant … Read more

If f:U1→Lp(μ;E2)f:U_1\to\mathcal L^p(\mu;E_2) is Fréchet differentiable, can we say anything about the Fréchet differentiability of u↦f(u)(ω)u\mapsto f(u)(\omega)?

Let (Ω,A,μ) be a σ-finite measure space, p≥1, Ei be a R-Banach space, U1⊆E1 be open and f:U1→L be Fréchet differentiable at x∈U1, where L=Lp(μ;E) or L=Lp(μ;E). For a function f of this form, can we say anything about the Fréchet differentiability of U1→E2,u↦f(u)(ω) for ω∈Ω (outside a μ-null set)? Strictly speaking, the notion of … Read more