International Conference on

“Nonlinear Evolution Equations”

Roma, January 28-31, 2003

On a phase transition model of Penrose–Fife type

Gianni Gilardi

Dipartimento di Matematica “F. Casorati”

Università degli Studi di Pavia Via Ferrata 1, 27100 Pavia, Italy

Tel:  +39 0382 50 56 42
Fax:  +39 0382 50 56 02
E-mail:  gilardi@dimat.unipv.it
Home page: http://www-dimat.unipv.it/˜ gilardi/

1 Introduction Two-phase Stefan problem (x ∈ Ω ⊂ ℝ^{3}, t ∈ (0, T))

• ∂_{t}e + div q = g  energy balance
e = e(x, t)  internal energy
q = q(x, t)  heat flux
g = g(x, t)  heat source
e = ϑ + λχ  constitutive law for e
q = −∇ϑ  Fourier law
ϑ = ϑ_{abs} − ϑ_{c} = ϑ(x, t) relative temperature
χ = χ(x, t)  phase parameter
λ = cnst> 0  latent heat
• χ ∈ H(ϑ)  χ = proportion of the solide phase

where H is the Heaviside graph

χ

1

0  ϑ

2 Introduction Typical phase field model

First note that

χ ∈ H(ϑ) ⟺ H^{−1}(χ) ∋ λϑ ^{λϑ ; χ}

Then relax (μ, ν > 0 small constants). Phase relaxation (Visintin 1985)

μ∂_{t}χ + H^{−1}(χ) ∋ λϑ

Phase field

μ∂_{t}χ − νΔχ + H^{−1}(χ) ∋ λϑ

Allen–Cahn dynamics

μ∂_{t}χ − νΔχ + W′(χ) = λϑ

where W is a double well potential like W(χ) = χ^{2}(1 − χ)^{2}.

Both phase field models are included in

∂_{t}e + div q = g
e = ϑ + λχ, q = −∇ϑ
μ∂_{t}χ − νΔχ + ∂j(χ) + σ′(χ) ∋ λϑ

where

j : ℝ → [0, +∞] convex, proper, l.s.c.
σ : ℝ → ℝ  smooth, σ′ Lipschitz

Phase dynamics is the gradient flow governed by the free energy functional

F_{ϑ}(χ) = \frac{ν}{2} ∫_{Ω} |∇χ|^{2} + ∫_{Ω}(j(χ) + σ(χ) − λϑχ)

Very wide literature, see lots of references in

M. Brokate & J. Sprekels: Hysteresis and phase transitions Springer-Verlag, 1996

A. Visintin:

Models of phase transitions Birkhäuser, 1996 Trouble of previous models

• linearization near ϑ_{abs} = ϑ_{c}

• no thermodynamical consistency

Penrose–Fife 1990

• no linearization

• thermodynamical consistency

• ϑ = ϑ_{abs} = absolute temperature: ϑ > 0

Generalized model

∂_{t}e + div q = g e = ϑ + λχ some constitutive law for q

μ∂_{t}χ − νΔχ + ∂j(χ) + σ′(χ) ∋ \frac{λ}{ϑ_{c}} − \frac{λ}{ϑ}

where j and σ as above • particular cases

• modified problems

• choice of the boundary conditions

Well-posedness Colli, Horn, Kenmochi, Laurençot, Niezgódka, Sprekels, Zheng, … (several combinations)

Other directions: long-time behavior, asymptotic analyses, memory, numerical approach, Cahn-Hilliard dynamics… 6 References

Zheng 1992: well-posedness Kenmochi, Niezgódka 1993 and 1994: well-posedness Horn 1993: numerical scheme Sprekels 1993: well-posedness Sprekels, Zheng 1993: well-posedness Kenmochi, Niezgódka 1994: well-posedness Laurençot 1994: well-posedness Colli, Sprekels 1995: asymptotic analysis Laurençot 1995: well-posedness Horn, Laurençot, Sprekels 1996: well-posedness Novick-Cohen, Zheng 1996: stationary solutions Klein 1998: time discretization Colli, Laurençot 1998: well-posedness Colli, Sprekels 1998: memory Krejčí, Sprekels 1998: connections with hysteresis and ν=0 Shirakawa, Ito, Yamazaki, Kenmochi 1998: well-posedness and long

time behavior

Laurençot 1998: well-posedness with Fourier law Colli, Laurençot, Sprekels 1999: well-posedness Colli, Sprekels 1999: well-posedness with Fourier law and ν=0 Kenmochi, Kubo 1999: well-posedness Klein 1999: time discretization Ito, Kenmochi 2000: inertial sets Stefanelli 2001: time discretization Colli, Grasselli, Ito 2002: parabolic-hyperbolic related problems Recupero 2002: memory Shen, Zheng 2002: actractors Ito, Kenmochi, Kubo (to appear): well-posedness, also C-H dyn Rocca (to appear): well-posedness with C-H dyn and memory Rocca, Schimperna (to appear): C-H dyn and Fourier law,

well-posedness

Rocca, Schimperna (to appear): C-H dyn and memory,

asymptotics

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References 6 • Colli–Laurençot 1998 (CL)

• Colli–Laurençot–Sprekels 1999 (CLS)

The constitutive law for q has the form

q = −∇α(ϑ)

where

α : (0, +∞) → ℝ

is strictly monotone, concave, onto suitable behavior near 0 and +∞ (thermodynamical consistency ensured)

Equivalent forms

q = −k_{1}(ϑ)∇ϑ  and  q = k_{2}(ϑ)∇(1/ϑ)

where

k_{1}(ϑ) = α′(ϑ) and k_{2}(ϑ) = ϑ^{2}α′(ϑ)

8 CL and CLS Take λ = μ = ν = 1 and set β := ∂j. Then for a new σ the problem reads

∂_{t}(ϑ + χ) − Δu = g

u = α(ϑ)

∂_{t}χ − Δχ + ξ + σ′(χ) = −\frac{1}{ϑ}

ξ ∈ β(χ)

Boundary conditions:

∂_{n}u + cu = given (third type b.c., c > 0)
∂_{n}χ = 0  (homogeneous Neumann b.c.)

Initial conditions for ϑ e χ.

CL: existence with

a very general α

CLS: regularity and

uniqueness among smooth sol’s with a much more particular α 9 CL and CLS Assumptions of CL: (besides strict monotonicity and concavity) here any a > 0

α(ϑ) ≈ −1/ϑ^{a} or α(ϑ) ≈ ln ϑ (ϑ → 0^{+})
α(ϑ) “between” ln ϑ and ϑ  (ϑ → +∞)

Assumptions of CLS:

\begin{matrix}α(ϑ) ≈ −1/ϑ & (ϑ → 0^{+}) \\ α(ϑ) ≈ ϑ & (ϑ → +∞) \end{matrix}

Joint paper with

Andrea Marson (Padova)

Math. Meth. Appl. Sci. 2003

• assumptions on the structure as general as possible (fill the gap, if possible)

• Dirichlet conditions for u corresponding to a given ϑ_{abs} > 0 on the bdry as a limit case

Assumptions between CL and CLS:

α(ϑ) ≈ −1/ϑ  (ϑ → 0^{+})
α(ϑ) “between” ln ϑ and ϑ (ϑ → +∞)

• Existence (third type converges to Dirichlet)

• Regularity and uniqueness among smooth sol’s (also for third type b.c.: CLS improved)

11 GM Convergence and existence

• penalize the Dirichlet condition

• the penalized problem is a third type b.v.p.

• a priori estimates and weak compactness

• strong compactness and usual monotonicity methods

Regularity and uniqueness

• CLS argument with modification

Mathematical tools

• use of elementary properties of α 12 GM Precise assumptions on α (besides strict monotonicity and concavity)

r^{2}α′(r) = 1 + o(1) as r → 0^{+}
r^{d}α′(r) = c_{∞} + o(1) as r → +∞ (c_{∞} > 0)
with 0 ≤ d ≤ 1 (d ≈ distance from Fourier)

Notation (r > 0)

α(r) = −\frac{1}{r} + ℓ(r) (ℓ = remainder)

• (α(r) − α(s)) (\frac{1}{s} − \frac{1}{r}) ≥ c (\frac{1}{s} − \frac{1}{r})^{2} c > 0

• ℓ^{2}(r) ≤ δα^{2}(r) + c_{δ}(1 + α̂(r))  (α′ = α)

• The function ℓ∘α^{−1} is Lipschitz continuous in ℝ • If o(1) becomes O(r) as r → 0^{+}, then

|(ℓ ∘ α^{−1})′(s)| ≤ c\sqrt{(α^{−1})′(s)} 13 GM e.b. ⟨∂_{t}(ϑ_{ε}(t) + χ_{ε}(t)), v⟩ + ∫_{Ω} ∇u_{ε}(t) · ∇v

+ \frac{1}{ε} ∫_{Γ}(u_{ε}(t) − u_{Γ}(t))v

= ∫_{Ω} g(t)v

∀ admissible v, a.e. in (0, T)

ph.d. ∂_{t}χ_{ε} − Δχ_{ε} + ξ_{ε} + σ′(χ_{ε}) = −\frac{1}{ϑ_{ε}}

Neumann b.c. for χ ϑ_{ε}(0) = ϑ_{0}, χ_{ε}(0) = χ_{0} 14 CGRS Joint paper with

P. Colli, E. Rocca, G. Schimperna (Pavia) in preparation

• assumptions as general as possible

• Neumann conditions for u

Assumptions on α as above

α(ϑ) ≈ −1/ϑ  (ϑ → 0^{+})
α(ϑ) “between” ln ϑ_{{↑ \atop d=1}} and ϑ_{{↑ \atop d=0}} (ϑ → +∞)

Further assumption, unfortunately

D(β) = ℝ and growth conditions at infinity in particular, no constraints on χ • Existence (third type converges to Neumann)

• Improvement of the uniqueness proof:

uniqueness of the nonsmooth solution (it works also for other b.c.)

• Regularity and uniqueness among smooth sol’s (as in GM)

Convergence and existence

• approximate the Neumann condition with third type b.c.

• a priori estimates and weak compactness

• strong compactness and usual monotonicity methods

• tricky point: main a priori estimate

New uniqueness proof

• modification of previous arguments

• assumptions on ℓ furtherly reinforced 16 CGRS Idea for the main a priori estimate Recall

r^{2}α′(r) = 1 + o(1) as r → 0^{+}

α(r) = −\frac{1}{r} + ℓ(r)

ℓ(r) = o(1/r)  as r → 0^{+}
r^{d}α′(r) = c_{∞} + o(1) as r → +∞

0 ≤ d ≤ 1

Ω ⊂ ℝ^{3} open, bounded, connected, smooth Γ = ∂Ω, Q = Ω × (0, T) V = H^{1}(Ω) ↪ H = L^{2}(Ω) ↪ V′ H_{n}^{2} = {v ∈ H^{2}(Ω) : ∂_{n}v = 0} reasonable assumptions on data

ϑ_{ε} ∈ L^{∞}(0, T; H) ∩ H^{1}(0, T; V′) ∩ …
χ_{ε} ∈ L^{2}(0, T; H_{n}^{2}) ∩ H^{1}(0, T; H)
u_{ε} ∈ L^{2}(0, T; V), ξ_{ε} ∈ L^{2}(Q)
ϑ_{ε} > 0 a.e. in Q, 1/ϑ_{ε} ∈ L^{2}(0, T; V)
u_{ε} = α(ϑ_{ε}), ξ_{ε} ∈ β(χ_{ε}) a.e. in Q

e.b. ⟨∂_{t}(ϑ_{ε}(t) + χ_{ε}(t)), v⟩ + ∫_{Ω} ∇u_{ε}(t) · ∇v

+ ε ∫_{Γ} u_{ε}(t)v

= ∫_{Ω} g(t)v + ∫_{Γ} h(t)v ∀ v ∈ V, a.e. in (0, T)

ph.d. ∂_{t}χ_{ε} − Δχ_{ε} + ξ_{ε} + σ′(χ_{ε}) = −\frac{1}{ϑ_{ε}} a.e. in Q

ϑ_{ε}(0) = ϑ_{0}, χ_{ε}(0) = χ_{0}

Limit problem: same regularity and similar equations with ε = 0 in (e.b.)

∫_{0}^{t} (e.b.) t = s, v = ϑ_{ε}(s) + u_{ε}(s) − cnst ds

+ ∫_{Q_{t}} (ph.d.) × ∂_{t}χ_{ε} dx ds

where Q_{t} = Ω × (0, t).

• main terms (lhs)

\frac{1}{2} ∫_{Ω} |ϑ_{ε}(t)|^{2} + ∫_{Ω} α̂(ϑ(t)) (α′ = α)

+ ∫_{Q_{t}} ∇u_{ε} · ∇ϑ_{ε} + ∫_{Q_{t}} |∇u_{ε}|^{2}

+ ∫_{Q_{t}} |∂_{t}χ_{ε}|^{2} + \frac{1}{2} ∫_{Ω} |∇χ_{ε}(t)|^{2} + ∫_{Ω} j(χ_{ε}(t))

• to be compensated (don’t worry)

∫_{Q_{t}} ∂_{t}χ_{ε} u_{ε} (lhs)  and  − ∫_{Q_{t}} \frac{∂_{t}χ_{ε}}{ϑ_{ε}} (rhs)

• “source” terms and “easy” terms (rhs)

19 CGRS • Source terms on the right hand side

∫_{Q_{t}} gv + ∫_{Γ×(0,t)} hv (v = ϑ_{ε} + u_{ε} − cnst) trouble !!!

Trouble 1

∫_{Γ×(0,t)} hϑ_{ε}

The left hand side can help just with

\begin{matrix}\frac{1}{2} ∫_{Ω} |ϑ_{ε}(t)|^{2} + ∫_{Ω} α̂(ϑ(t)) \\ + ∫_{Q_{t}} ∇u_{ε} · ∇ϑ_{ε} + ∫_{Q_{t}} |∇u_{ε}|^{2} \end{matrix}

Lemma. Assume

v ∈ L^{2}(Ω), v > 0, and ∇α(v) ∈ L^{2}(Ω)^{3}

and set

d_{•} = \frac{4}{1 + 3d} for 0 ≤ d ≤ 1 .

Then the trace of v belongs to L^{d_{•}}(Γ) and

∥v∥_{L^{d•} (Γ)} ≤ δ ∥∇α(v)∥_{H}^{2} + c_{δ} (1 + ∥v∥_{H}^{2})

for any δ > 0 and some c_{δ} = c(δ, Ω, α).

Proof. Use the Gagliardo trace theorem and the Hölder and Sobolev inequalities.

So, reasonable assumptions on h yield (h ∈ L^{∞}(Γ × (0, T)) works for any d ∈ [0, 1])

∫_{Γ×(0,t)} hϑ_{ε} ≤ δ ∫_{Q_{t}} |∇u_{ε}|^{2} + c_{δ} ∫_{Q_{t}} |ϑ_{ε}|^{2} + c_{δ}

and this can be controlled by the left hand side.

21 CGRS Trouble 2

∫_{Q_{t}} gu_{ε} + ∫_{Γ×(0,t)} hu_{ε} ≤ c(g, h) ∥u_{ε}∥_{L^{2}(0,T;V)} with the full V -norm, while the l.h.s. contains only the seminorm. So, we have to be careful. Setting for convenience

⟨f(t), v⟩ := ∫_{Ω} g(t)v + ∫_{Γ} h(t)v for v ∈ V our trouble becomes

∫_{0}^{t} ⟨f(s), u_{ε}(s)⟩ ds

We are going to use the Poincaré inequality

∥u_{ε}(t) − m_{ε}(t)∥_{V}^{2} ≤ c ∫_{Ω} |∇u_{ε}(t)|^{2} where m_{ε} is the mean value

m_{ε}(t) = ∫_{Ω} u_{ε}(t)

22 CGRS We split and estimate our integral this way

∫_{0}^{t} ⟨f(s), u_{ε}(s)⟩ ds

= ∫_{0}^{t} ⟨f(s), u_{ε}(s) − m_{ε}(s)⟩ ds
+ ∫_{0}^{t} m_{ε}(s) ⟨f(s), 1⟩ ds
≤ δ ∫_{Q_{t}} |∇u_{ε}|^{2} + c_{δ} ∥f∥_{L^{2}(0,T;V′)}^{2}
+ ∥f∥_{L^{∞}(0,T;V′)} ∥1∥_{V} ∫_{0}^{t} |m_{ε}(s)| ds

(even better…) We have to estimate the last integral We split the mean value this way

\begin{matrix}m_{ε}(t) = ∫_{Ω} u_{ε}(t) = ∫_{Ω} α(ϑ_{ε}(t)) \\ = − ∫_{Ω} \frac{1}{ϑ_{ε}(t)} + ∫_{Ω} ℓ(ϑ_{ε}(t)) \end{matrix}

Hence (forget the remainder, please)

∫_{0}^{t} |m_{ε}(s)| ds ≤ ∫_{0}^{t} ( ∫_{Ω} \frac{1}{ϑ_{ε}(s)}) ds + …

The last mean value is computed via (ph.d.)

−\frac{1}{ϑ_{ε}} = ∂_{t}χ_{ε} − Δχ_{ε} + ξ_{ε} + σ′(χ_{ε})

24 CGRS Take the mean value and use ∂_{n}χ = 0

∫_{Ω} \frac{1}{ϑ_{ε}} ≤ c ∫_{Ω} |∂_{t}χ_{ε}| + c ∫_{Ω} |ξ_{ε}| + c ∫_{Ω}(1 + |χ_{ε}|)

Finally integrate over (0, t)

∫_{0}^{t} ( ∫_{Ω} \frac{1}{ϑ_{ε}(s)}) ds

≤ δ ∫_{Q_{t}} |∂_{t}χ_{ε}|^{2} + c_{δ} + c ∫_{Q_{t}}(1 + |χ_{ε}|^{2}) + c ∫_{Q_{t}} |ξ_{ε}| (recall ξ_{ε} ∈ β(χ_{ε}))

Everything works but the last integral. It should be controlled by the term

∫_{Ω} j(χ_{ε}(t))

on the l.h.s. We need the further assumption

|s| ≤ c (1 + j(r)) ∀ r ∈ ℝ ∀ s ∈ β(r) in order to use Gronwall.

CGRS 24 25 CGRS Uniqueness proof

e.b. ⟨∂_{t}(ϑ(t) + χ(t)), v⟩ + ∫_{Ω} ∇u_{ε}(t) · ∇v

= ∫_{Ω} g(t)v + ∫_{Γ} h(t)v ∀ v ∈ V, a.e. in (0, T)

ph.d. ∂_{t}χ − Δχ + ξ + σ′(χ) = −\frac{1}{ϑ} a.e. in Q ϑ(0) = ϑ_{0}, χ(0) = χ_{0} Integrate the e.b. in t

i.e.b. ∫_{Ω}(ϑ + χ)v + ∫_{Ω} ∇ (∫_{0}^{t} u) · ∇v = ⟨known, v⟩ Write eqn’s for two solution and take the difference. I set ϑ = ϑ_{1} − ϑ_{2}, etc., or write diff{…}. Then

δ i.e.b. ∫_{Ω} ϑv + ∫_{Ω} χv + ∫_{Ω} ∇ (∫_{0}^{t} u) · ∇v = 0 δ ph.d. ∂_{t}χ − Δχ + ξ + diff{σ′(χ_{i})} = diff{−1/ϑ_{i}} 26 CGRS Now

∫_{0}^{t} (δ i.e.b.)(s) v = u(s) ds + ∫_{Q_{t}} (δ ph.d.) × χ We obtain

∫_{Q_{t}} ϑ u + ∫_{Q_{t}} χu + \frac{1}{2} ∫_{Ω} |∇ ∫_{0}^{t} u|^{2}

+ \frac{1}{2} ∫_{Ω} |χ(t)|^{2} + ∫_{Q_{t}} |∇χ|^{2} + ∫_{Q_{t}} ξχ

= ∫_{Q_{t}} diff{−1/ϑ_{i}}χ − easy term Two possibilities

use u and write ϑ_{i} = α^{−1}(u_{i})
use ϑ and write u_{i} = α(ϑ_{i})

Previous argument (CLS and GM): use u. This leeds to the integral

∫_{Q_{t}}(1 + u_{1}^{2} + u_{2}^{2})χ^{2}

Playing with Hölder, one sees that Gronwall works with smooth u_{i}, namely

u_{i} ∈ L^{∞}(0, T; L^{6}(Ω))

Such a smoothness needs a regularity result. Assuming just the first reinforcement

r^{2}α′(r) = 1 + O(r) as r → 0^{+}

and the data to be smoother, one proves

u_{i} ∈ L^{∞}(0, T; V)

and uses the 3D Sobolev inclusion V ⊆ L^{6}(Ω)

28 CGRS Use ϑ, instead Recall

u_{i} = α(ϑ_{i}) and α(r) = −\frac{1}{r} + ℓ(r)

Reinforce a little more, namely

ℓ Lipschitz continuous near 0

whence ℓ globally Lipschitz continuous.

∫_{Q_{t}} ϑ u + \underbrace{∫_{Q_{t}} χu}_{(1)}

+ \frac{1}{2} ∫_{Ω} |χ(t)|^{2} + ∫_{Q_{t}} |∇χ|^{2} + …

= \underbrace{∫_{Q_{t}} diff\{−1/ϑ_{i}\}χ}_{(2)} + …

Compensate (1) and (2) and use Lipschitz 29 CGRS

∫_{Q_{t}} ϑ u + \frac{1}{2} ∫_{Ω} |χ(t)|^{2} + ∫_{Q_{t}} |∇χ|^{2}

≤ ∫_{Q_{t}} | diff{ℓ(ϑ_{i})}||χ| + … ≤ c ∫_{Q_{t}} |ϑ||χ| + …

We have

ϑu = ϑ diff\{α(ϑ_{i})\} ≥ δ_{0} \frac{ϑ^{2}}{1 + ϑ_{1}^{d} + ϑ_{2}^{d}}

Hence we use the elementary inequality

|ϑ||χ| ≤ \frac{δ_{0}}{2} \frac{ϑ^{2}}{1 + ϑ_{1}^{d} + ϑ_{2}^{d}} + \frac{1}{2δ_{0}} (1 + ϑ_{1}^{d} + ϑ_{2}^{d})χ^{2}

and have

\begin{matrix}\frac{δ_{0}}{2} ∫_{Q_{t}} \frac{ϑ^{2}}{1 + ϑ_{1}^{d} + ϑ_{2}^{d}} + \frac{1}{2} ∥χ(t)∥_{H}^{2} + ∫_{Q_{t}} |∇χ|^{2} \\ ≤ c ∫_{Q_{t}}(1 + ϑ_{1}^{d} + ϑ_{2}^{d})χ^{2} \end{matrix}

Then OK if d = 0. Assume 0 < d ≤ 1. We still want to apply Gronwall. CGRS 29 30 CGRS Write

∫_{Q_{t}} ϑ_{i}^{d}χ^{2} = ∫_{0}^{t} ∫_{Ω} ϑ_{i}^{d}χχ

and play with Hölder

\begin{matrix}p, q ≥ 1 and \frac{1}{p} + \frac{2}{q} = 1 \\ p := \frac{2}{d} ∈ [2, ∞) and q := \frac{4}{2 − d} ∈ (2, 4] \end{matrix}

Then

\begin{matrix}∫_{Q_{t}} ϑ_{i}^{d}χ^{2} ≤ ∫_{0}^{t} ∥ϑ_{i}^{d}∥_{L^{p}(Ω)} ∥χ∥_{L^{q}(Ω)}^{2} \\ = ∫_{0}^{t} ∥ϑ_{i}∥_{H}^{d} ∥χ∥_{L^{q}(Ω)}^{2} ≤ c ∫_{0}^{t} ∥χ∥_{L^{4}(Ω)}^{2} \end{matrix}

since ϑ_{i} ∈ L^{∞}(0, T; H) and q ≤ 4. The compact embedding V ⊂ L^{4}(Ω) yields

∥v∥_{L^{4}} ≤ δ ∥∇v∥_{H} + c_{δ} ∥v∥_{H}

and we obtain

∫_{Q_{t}}(ϑ_{1}^{d} + ϑ_{2}^{d})χ^{2} ≤ \frac{1}{2} ∫_{Q_{t}} |∇χ|^{2} + C ∫_{0}^{t} ∥χ∥_{H}^{2}

Cahn–Hilliard type dynamics ∂_{t}(ϑ + χ) − Δα(ϑ) = g ∂_{t}χ − Δw = 0

w ∈ −Δχ + ∂j(χ) + σ′(χ) − \frac{1}{ϑ_{c}} + \frac{1}{ϑ}

homogeneous Neumann b.c. for χ and w some b.c. for either ϑ or u initial conditions for ϑ and χ

with α as before (essentially)

• Main difference in the a priori estimates some ∥ · ∥_{H} replaced by ∥ · ∥_{V′}

• existence result for the third type problem for u (0 ≤ d < 1, with A. Marson)

• existence for the Dirichlet problem should work as in GM

• uniqueness works as before ∥v∥_{L^{4}} ≤ δ ∥∇v∥_{H} + c_{δ} ∥v∥_{V′}

• dealing with Neumann conditions… (hope)