Book TE-TEC

1.1 Thermoelectrical coupled systems . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Formulation of the TE problem . . . . . . . . . . . . . . . . . . . . . .2

1.1.2 Radiative effects on the thermoelectric problems . . . . . . . .4

1.2 Thermoelectrochemical coupled systems . . . . . . . . . . . . . .  . . . . . 6

1.2.1 Constitutive and governing equations . . . . . . . . . . . . . . . . .7

1.2.2 Onsager reciprocity relations . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.4 Formulation of the TEC problem . . . . . . . . . . . . . . . . . . . .13

2.1 Preliminaries of functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Sobolev and other related spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 

2.3 Miscelanea of existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Steady-state regularity on the Dirichlet problem . . . . . . . . . . . . . . . . .28

2.4.1 L^^{p}-regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.2 W^{s,p}-regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Steady-state regularity on the Neumann problem . . . . . . . . . . . . . . . 32

2.6 Steady-state regularity on the Zaremba problem . . . . . . . . . . . . . . . .35

2.7 W^{1,p} regularity on the mixed Neumann-Robin problem . . . . . . . . 38

2.8 Maximal parabolic regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41

2.9 Parabolic regularity on the Dirichlet problem . . . . . . . . . . . . . . . . . . . 42

2.10 W^{1,0}_p (Q_T ) regularity (isotropic case) . . . . . . . . . . . . . . . . . . .48

3.1 The Joule-Peltier-Seebeck-Thomson effect . . . . . . . . . . . . . . . . . .51

3.2 Data assumptions and main existence results . . . . . . . . . . . . . . . .52

3.3 Existence result to an auxiliary electric problem . . . . . . . . . . . . . . 57

3.4 Existence results to auxiliary thermic problems . . . . . . . . . . . . . . .60

3.5 Spatio-independent Seebeck coefficient (proof of Theorem 3.1) . .70

3.6 Regularity results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74

3.6.1 Proof of Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . .75

3.6.2 Proof of Proposition 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . .77

3.7 Spatio-dependent Seebeck coefficient (proof of Theorem 3.2). . .78

3.7.1 The validation of the existence of the radius R . . . . . . . . 78

3.7.2 The weak sequential continuity of T. . . . . . . . . . . . . . . . . 79

3.8 2D existence result (proof of Theorem 3.3) . . . . . . . . . . . . . . . . . 80

3.9 A limit model for thermoelectric equations . . . . . . . . . . . . . . . . . . 81. 

3.9.1 Proof of Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.1 Statement of the TEC problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.2 Data assumptions and main existence results . . . . . . . . . . . . . . . . . . .86

4.3 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.4 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.5 Minimum principle (proof of Proposition 4.1) . . . . . . . . . . . . . . . . . . . 100

5.1 Statement of the problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2 Mixed Neumann-Dirichlet elliptic problems . . . . . . . . . . . . . . . .103

5.2.1 H^1-solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104

5.2.2 W^{1,q}-solvability (q ≤n/(n−1)) . . . . . . . . . . . . . . . . . . .105

5.2.3 L∞-estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109

5.3 Mixed Neumann-Robin-type elliptic problems . . . . . . . . . . . . . 115

5.3.1 V^{2,ℓ}-solvability (ℓ ≥2) . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.3.2 L^q-estimates (1/q > 1/p−1/[2(n−1)], 2 < p < 2(n−1)) . . . .118

5.3.3 L^∞-estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121

5.3.4 V_{q,ℓ−1}-solvability (q < n/(n−1), ℓ ≥2). . . . . . . . . . . . . . . 128

5.3.5 Robin-Neumann problem (ℓ= 2) . . . . . . . . . . . . . . . . . . . . 129

5.4 Potential theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .130

5.4.1 Green kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132

5.4.2 W^{1,p}-estimates (p > n). . . . . . . . . . . . . . . . . . . . . . . . . 140

5.4.3 W^{1,q}-estimate (q < n/(n−1)) for the Green kernel . . . . 142

6.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.1.1 General boundary operator γ of the power-type . . . . . . . . .146

6.2 L^{p,∞}(Q_T ) and L^{ℓ+p−2}(Σ_T ) estimates . . . . . . . . . . . . . . .147

6.3 Main results for quasi-linear boundary operator γ. . . . . . . . . . . . 149

6.4 Proof of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154

6.4.1 Proof of the estimates (6.16)-(6.17) for um. . . . . . . . . . .154

6.4.2 Passage to the limit in (6.28) as m →∞. . . . . . . . . . . . . .155

6.5 Minimum and maximum principles . . . . . . . . . . . . . . . . . . . . . . . .157

6.6 Proof of Theorem 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.6.1 L^{1,∞}(Q_T )- and L^{ℓ−1}(Σ_T )- estimates (6.21) for u_m. . 162

6.6.2 L^p-estimate (6.22) to the gradient of u_m. . . . . . . . . . . . . . .  162

6.6.3 Estimate of ∂_t u_m in L1(0, T ; W^{1,p′}(Ω)). . . . . . . . . . . . . .164

6.6.4 Passage to the limit in (6.28) as m →∞. . . . . . . . . . . . . . . . . . 165

6.7 The case of b_# = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.7.1 Proof of Theorem 6.4 . . . . . . . . . . . . . . . . . . . . . . . . 165

6.7.2 Proof of Theorem 6.5 . . . . . . . . . . . . . . . . . . . . . . . . 166

7.1 Reverse H"older inequalities with increasing supports . . . . . . . . . . . .169

7.1.1 The stationary case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

7.1.2 The non-stationary case . . . . . . . . . . . . . . . . . . . . . . . . . . . .178

7.2 Quantative estimates (stationary case) . . . . . . . . . . . . . . . . . . . . . . . .184

7.3 Local higher regularity of the gradient . . . . . . . . . . . . . . . . . . . . . . . . .187

7.3.1 Interior higher regularity of the gradient . . . . . . . . . . . . . . . . 187

7.3.2 Higher regularity up to the boundary of the gradient . . . . . . .189

7.4 Steady-state higher-regularity results . . . . . . . . . . . . . . . . . . . . . . . . . 191

7.4.1 Proof of Theorem 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191

7.4.2 Proof of Theorem 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192

7.5 Quantative estimates (non-stationary case) . . . . . . . . . . . . . . . . . . . . 193

7.6 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

7.7 Higher regularity on a Cauchy-Neumann-power type problem (proof of Theorem 7.3) . .. 196

7.7.1 Local interior higher integrability of the gradient . . . . . . . . . .  . . . . .  . . . . . 196

7.7.2 Local higher integrability up to the spatial boundary of the gradient . . . . . . 197

7.7.3 Global higher integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . 199

8.1 Statement of the TE variational problem . . . . . . . . . . . . . . . . . . . . . . . 201

8.2 Data assumptions and the main existence result . . . . . . . . . . . . . . . . .202

8.3 W^{1,p}-existence (proof of Theorem 8.1) . . . . . . . . . . . . . . . . . . . . . . 203

9.1 Statement of the TEC variational problem . . . . . . . . . . . . . . . . . . . . . . 207

9.2 Data assumptions and the main existence result . . . . . . . . . . . . . . . . .209

9.3 Strategy of the proof of Theorem 9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 211

9.4 Existence of auxiliary solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .212

9.5 Existence result (proof of Theorem 9.1) . . . . . . . . . . . . . . . . . . . . . . . . 217

9.6 Example applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221