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The uniqueness of the solution of a nonlinear heat conduction problem under Hölder’s continuity condition
Applied Mathematics Letters 103 (2020) 106214
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
The uniqueness of the solution of a nonlinear heat conduction problem under Hölder’s continuity condition Michal Křížek Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ-115 67 Prague 1, Czech Republic
article
info
abstract We investigate a stationary nonlinear heat conduction problem in which heat conductivities depend on temperature. It is known that such problem need not have a unique solution even when the conductivity coefficients are continuous. In this paper we prove that for 12 -Hölder continuous coefficients the uniqueness of the weak solution is guaranteed. © 2020 Published by Elsevier Ltd.
Article history: Received 20 September 2019 Received in revised form 31 December 2019 Accepted 31 December 2019 Available online 8 January 2020 Dedicated to Dr. Milan Práger on his 90th birthday Keywords: Weak solution Nonlinear heat conduction Heat transfer coefficient Hölder continuity
1. Introduction The problem of a stationary heat conduction in nonhomogeneous, anisotropic, and nonlinear media with mixed boundary conditions consists of finding u ∈ C 1 (Ω) ∩ C 2 (Ω) such that −div(A(x, u) grad u) = F (x, u) in
Ω,
(1)
on
Γ1 ,
(2)
n A(s, u) grad u = G(s, u) on
Γ2 ,
(3)
u =u ⊤
where Ω ⊂ Rd , d ∈ {1, 2, . . . }, is a bounded domain with Lipschitz continuous boundary ∂Ω, n is the outward unit normal to ∂Ω, Γ1 , Γ2 ⊂ ∂Ω are relatively open sets with respect to the topology on ∂Ω such that Γ1 ∩ Γ2 = ∅, Γ 1 ∪ Γ 2 = ∂Ω, u is the unknown temperature, and u ∈ W21 (Ω) is a given temperature. Further, we assume that A = (Aij )di,j=1 is a measurable matrix function of heat conductivities such that ess supx,ξ,i,j |Aij (x, ξ)| ≤ C. Suppose that there exist positive constants CE and CH such that for almost all x ∈ Ω we have η ⊤ A(x, ξ)η ≥ CE ∥η∥2 ∀ξ ∈ R1 ∀η ∈ Rd (4) E-mail address: [email protected]. https://doi.org/10.1016/j.aml.2020.106214 0893-9659/© 2020 Published by Elsevier Ltd.
2
M. Křížek / Applied Mathematics Letters 103 (2020) 106214
and α
|Aij (x, ζ) − Aij (x, ξ)| ≤ CH |ζ − ξ| ∀ζ, ξ ∈ R1 , i, j = 1, . . . , d, (5) [1 ] where ∥ · ∥ stands for the Euclidean norm and α ∈ 2 , 1 is a given H¨older exponent, i.e., A(·, ·) is α-H¨older continuous with respect to the last variable. Finally, F (·, v(·)) ∈ L2 (Ω) and G(·, v(·)) ∈ L2 (Γ2 ) for any v ∈ W21 (Ω), and let for almost all x ∈ Ω and almost all s ∈ Γ2 the functions v ∈ R1 ↦→ F (x, v) ∈ R1
and v ∈ R1 ↦→ G(s, v) ∈ R1
be nonincreasing.
(6)
In particular, these functions can be independent of v. The function F may have, for example, the following frequently used form F (x, v) = f (x) − c(x)v,
x ∈ Ω,
where f is the density of volume heat sources and c ≥ 0. Similarly, we can set G(s, v) = g(s) − h(s)v for s ∈ Γ2 , where g is the density of surface heat sources and h ≥ 0 is the heat transfer coefficient. Several real-life applications of the problem (1)–(3) are given in [1]. In [2] we proved the uniqueness of the classical solution of the above problem (1)–(3) only for a diagonal matrix A and for Lipschitz continuous heat conductivities Aij with respect to the last variable, Aij (x, ·) ∈ 1 (R) for almost all x ∈ Ω, i.e., when α = 1 in inequality (5). This technique was generalized in [3] to W∞ prove the uniqueness of the weak solution of (1)–(3). Note that we could not use the theory of monotone operators, since the operator corresponding to (1)–(3) is nonmonotone, see [3]. Moreover, this operator is nonpotential, which means that the problem (1)–(3) cannot be transformed to the minimization of some functional [1]. If an elliptic equation is not in the divergence form there exist examples of non-unique solutions, see [4, p. 209], [5]. Also semiconductor elliptic equations (describing e.g. thyristor) may have several different solutions for certain fixed boundary conditions, see [6, p. 43] and [7]. In [3, p. 182] we present a simple one-dimensional example of nonuniqueness of the weak solution of the stationary heat conduction problem (1)–(3), when A is continuous, but not Lipschitz continuous. This example shows that there exist infinitely many solutions for any kind of boundary conditions. In this paper we show that the uniqueness of the weak solution can be guaranteed also for 21 -H¨ older continuous heat conductivities. 2. Main uniqueness theorem Throughout this paper we shall use the standard Sobolev space notation, (·, ·)0,Ω and ⟨·, ·⟩0,Γ2 will denote the usual scalar product in (L2 (Ω))m , m = 1, 2, . . . , and L2 (Γ2 ), respectively. For simplicity, in what follows we shall often write only A(u) instead of A(x, u(x)), F (u) instead of F (x, u(x)), G(u) instead of G(s, u(s)), i.e., we shall not explicitly indicate the space variable x and the boundary variable s. Set V = {v ∈ W21 (Ω) | v = 0 on Γ1 }. Definition 1. A function u ∈ W21 (Ω) is called a weak solution of the problem (1)–(3) if u − u ∈ V, (A(u)grad u, grad v)0,Ω = (F (u), v)0,Ω + ⟨G(u), v⟩0,Γ2
(7) ∀v ∈ V.
(8)
Theorem 1. Let (4), (5), and (6) hold, let u1 and u2 be two weak solutions of the problem (7)–(8), and let Γ1 be nonempty. Then u1 = u2 a.e. in Ω.
M. Křížek / Applied Mathematics Letters 103 (2020) 106214
3
Fig. 1. Schematic illustration of function values of vε over Ω .
Proof . Define the following set Ω0 = {x ∈ Ω | u2 (x) > u1 (x)} and suppose that the d-dimensional Lebesgue measure of this set is positive, i.e. measd Ω0 > 0.
(9)
Next we will show that this inequality does not hold. Let ε > 0 be arbitrary and put (see Fig. 1) Ωε = {x ∈ Ω0 | u2 − u1 > ε}.
(10)
We shall use a special test function in (8), namely, { min(ε, u2 − u1 ) in Ω0 , vε = 0 in Ω \ Ω0 .
(11)
Using (7), we find that u2 −u, u1 −u ∈ V and thus u2 −u1 ∈ V . Consequently, the positive part (u2 −u1 )+ lies also in V . The reason is that v ↦→ v + represents a continuous mapping from W21 (Ω) into itself, see [8, p. 29] or [9, p. 58]. The mapping v ↦→ |v| = v + + v − is continuous, too. Hence, the relation (11) and the equality 1 min(a, b) = (a + b − |a − b|) 2 yield that vε = min(ε, (u2 − u1 )+ ) ∈ W21 (Ω ) and vε ∈ V, (12) since Γ1 ∩ Ω0 = ∅. According to (8) and (12), we find that (A(u1 )grad u1 , grad vε )0,Ω =(A(u2 )grad u2 , grad vε )0,Ω + (F (u1 ) − F (u2 ), vε )0,Ω + ⟨G(u1 ) − G(u2 ), vε ⟩0,Γ2 .
(13)
Clearly vε ≥ 0 in Ω. Thus from (6) we obtain (F (u1 ) − F (u2 ))vε ≥ 0
on Ω,
(G(u1 ) − G(u2 ))vε ≥ 0
From (4), (11), (12), (13), and (14) we get ∫ ∫ CE ∥grad vε ∥2 dx ≤ (grad vε )⊤ A(u1 )grad vε dx Ω ∫Ω = (grad vε )⊤ A(u1 )grad (u2 − u1 )dx Ω0 \Ωε
on Γ2 .
(14)
M. Křížek / Applied Mathematics Letters 103 (2020) 106214
4
∫ =
(grad vε )⊤ (A(u1 )grad u2 − A(u1 )grad u1 )dx
∫Ω
(grad vε )⊤ (A(u1 )grad u2 − A(u2 )grad u2 )dx ∫ − (F (u1 ) − F (u2 ))vε dx − (G(u1 ) − G(u2 ))vε ds Ω Γ2 ∫ ≤ (grad vε )⊤ (A(u1 ) − A(u2 ))grad u2 dx.
=
Ω∫
(15)
Ω
The last integral can be estimated by means of (10), (11), the Cauchy–Schwarz inequality, and (5) as follows ∫ (grad vε )⊤ (A(u1 ) − A(u2 ))grad u2 dx Ω0 \Ωε (∫ )1/2 (∫ )1/2 2 ≤ ∥(A(u1 ) − A(u2 ))grad u2 ∥ dx ∥grad vε ∥2 dx Ω0 \Ωε Ω0 \Ωε (∫ )1/2 (16) ≤ εα CH d1/2 ∥grad vε ∥0,Ω0 \Ωε . ∥grad u2 ∥2 dx Ω0 \Ωε
From this and (15) we find that there exists a constant C0 > 0 such that ∥grad vε ∥0,Ω ≤ C0 εα ∥grad u2 ∥0,Ω0 \Ωε .
(17)
According to the generalized Friedrichs inequality, there exists a constant C1 (p) > 0 such that ∥v∥0,p ≤ C1 (p)∥grad v∥0,p
for p ≥ 1,
(18)
where ∥ · ∥0,p stands for the standard Lebesgue Lp -norm. Further, we shall use the imbedding (L2 (Ω))d ⊂ (Lp (Ω))d for p ∈ [1, 2], i.e., there exists a constant C2 (p) such that ∥v∥0,p ≤ C2 (p)∥v∥0,2
∀v ∈ L2 (Ω).
(19)
Now from relations (10), (12), (18), (19), and (17) squared with p = 2α [1
] for α ∈ 2 , 1 , we obtain
≤
∫
∫
p |vε | dx = ε−p ∥vε ∥p0,p ≤ ε−p C1p (p)∥grad vε ∥p0,p Ωε Ω ε−p C1p (p)C2 (p)∥grad vε ∥p0 ≤ C0 C1p (p)C2 (p)∥grad u2 ∥p0,Ω0 \Ωε → 0 as
measd Ωε = ε−p
εp dx ≤ ε−p
ε → 0,
(20)
where u2 ∈ V is fixed, grad vε ∈ (L2 (Ω))d , and the constant C0 C1p (p)C2 (p) > 0 is independent of ε. However, this contradicts (9) and (10), because measd Ωε → measd Ω0
as ε → 0.
Therefore, measd Ω0 = 0 and u1 ≥ u2 a.e. in Ω. Interchanging the role of u1 and u2 , we get u2 ≥ u1 a.e. in Ω, and thus the uniqueness follows. □ Remark 1. When p ∈ (0, 1), then the symbol ∥ · ∥0,p stands for the so-called quasi-norm which satisfies all norm axioms except for the usual triangle inequality which is replaced by ∥v + w∥0,p ≤ C3 (p)(∥v∥0,p + ∥w∥0,p ) with some constant C3 (p) > 1. A natural question arises whether the Friedrichs inequality (18) can be extended to the case p ∈ (0, 1).
M. Křížek / Applied Mathematics Letters 103 (2020) 106214
5
Remark 2. In [3], we were not able to prove uniqueness of the Galerkin solutions of (1)–(3), since the corresponding test function defined like in (12) does not belong to finite element spaces, in general. However, the uniqueness was proved quite recently in [10] by a different approach. The nonlinear discrete problem is usually transformed to a sequence of linear problems by the method of successive approximations (the freezing coefficient method), see [1]. 3. Further uniqueness theorems In this section, we prove similar statements as in Theorem 1 under different assumptions. Now the set Γ1 can be empty. Theorem 2. Let (4), (5), and (6) hold, let u1 and u2 be two weak solutions of the problem (7)–(8), and let there exist a constant C > 0 and a nonempty relatively open subset Γ3 ⊂ Γ2 such that v2 − v1 ≤ C(G(s, v1 ) − G(s, v2 )) for almost all s ∈ Γ3 and all v1 , v2 ∈ R1 , v2 ≥ v1 . Then u1 = u2 a.e. in Ω. Proof . Using (12), we get 0 ≤ vε ≤ (u2 − u1 )+ = u2 − u1 ≤ C(G(u1 ) − G(u2 ))
on
Γ3 ∩ Ω 0 .
Hence, vε2 ≤ C(G(u1 ) − G(u2 ))vε
on
Γ3 ,
(21)
since (see Fig. 1) vε = 0
on
Γ3 \ Ω 0 .
(22)
Applying the following Friedrichs inequality, (21), (22), (13), (4), the Cauchy–Schwarz inequality, (5), and the inequality (17), which is independent of particular boundary conditions, we get ∥vε ∥20,Ω ≤C1 (∥vε ∥20,Γ3 + ∥grad vε ∥20,Ω ) (∫ ) ≤C2 (G(u1 ) − G(u2 ))vε ds + ∥grad vε ∥20,Ω Γ3 (∫ ) ≤C3 (G(u1 ) − G(u2 ))vε ds + CE ∥grad vε ∥20,Ω Γ3 (∫ ≤C3 (grad vε )⊤ (A(u1 )grad u1 − A(u2 )grad u2 )dx Ω ∫ ∫ ) − (F (u1 ) − F (u2 ))vε dx + (grad vε )⊤ A(u1 )grad (u2 − u1 )dx Ω ∫ Ω ⊤ ≤C3 (grad vε ) (A(u1 ) − A(u2 ))grad u2 dx Ω0 \Ωε (∫ )1/2 α 1/2 ≤C3 ε CH d ∥grad u2 ∥2 dx ∥grad vε ∥0,Ω0 \Ωε Ω0 \Ωε ∫ ≤C4 ε2α ∥grad u2 ∥2 dx. Ω0 \Ωε
From this and (19) we obtain for p = 2α similarly to (20) that ∫ ∫ p measd Ωε ≤ ε−p |vε | dx ≤ C2 (p)ε−p ∥vε ∥20,Ω ≤ C4 (p) Ω
∥grad u2 ∥2 dx → 0
Ω0 \Ωε
which again leads to a contradiction like at the end of the proof of Theorem 1. □
as ε → 0,
(23)
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M. Křížek / Applied Mathematics Letters 103 (2020) 106214
Theorem 3. Let (4), (5), and (6) hold, let u1 and u2 be two weak solutions of the problem (7)–(8), and let ˜ ⊂ Ω such that there exist a constant C > 0 and a nonempty open subset Ω v2 − v1 ≤ C(F (x, v1 ) − F (x, v2 )) ˜ and all v1 , v2 ∈ R1 , v1 ≥ v2 . Then u1 = u2 a.e. in Ω. for almost all x ∈ Ω The proof is similar to that of Theorem 2. We only exchange the role of F and G and instead of the Friedrichs inequality (23) we will use the generalized Poincar´e inequality (see e.g. [9, p. 113]) ∥vε ∥20,Ω ≤ C(∥vε ∥20,Ω˜ + ∥grad vε ∥20,Ω ). Acknowledgments The author is indebted to Jan Brandts, Lawrence Somer, and Tom´aˇs Vejchodsk´ y for fruitful discussions. This paper was supported by grants 18-09628S and 20-01074S of the Grant Agency of the Czech Republic and RVO 67985840 of the Czech Republic. References [1] M. Kˇr´ıˇ zek, P. Neittaanm¨ aki, Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications, Kluwer Academic Publishers, Dordrecht, 1996. [2] I. Hlav´ aˇ cek, M. Kˇr´ıˇ zek, On a nonpotential and nonmonotone second order elliptic problem with mixed boundary conditions, Stability Appl. Anal. Contin. Media 3 (1993) 85–97. [3] I. Hlav´ aˇ cek, M. Kˇr´ıˇ zek, J. Mal´ y, On Galerkin approximations of a quasilinear nonpotential elliptic problem of a nonmonotone type, J. Math. Anal. Appl. 184 (1994) 168–189. [4] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1977. [5] N.G. Meyers, An example of non-uniqueness in the theory of quasi-linear elliptic equations of second order, Arch. Ration. Mech. Anal. 14 (1963) 177–179. [6] P.A. Markowich, The Stationary Semiconductor Device Equations, Springer-Verlag, Wien, 1986. [7] M.S. Mock, An example of nonuniqueness of stationary solutions in semiconductor device model, COMPEL 1 (1982) 165–174. [8] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, Berlin, 1977. [9] J. Neˇ cas, Direct Methods in the Theory of Elliptic Equations, in: Springer Monographs in Mathematics, Springer, Heidelberg, 2012. [10] S. Pollock, Y. Zhu, Uniqueness of discrete solutions of nonmonotone PDEs without a globally fine mesh condition, Numer. Math. 139 (2018) 845–865.
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
The uniqueness of the solution of a nonlinear heat conduction problem under Hölder’s continuity condition Michal Křížek Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ-115 67 Prague 1, Czech Republic
article
info
abstract We investigate a stationary nonlinear heat conduction problem in which heat conductivities depend on temperature. It is known that such problem need not have a unique solution even when the conductivity coefficients are continuous. In this paper we prove that for 12 -Hölder continuous coefficients the uniqueness of the weak solution is guaranteed. © 2020 Published by Elsevier Ltd.
Article history: Received 20 September 2019 Received in revised form 31 December 2019 Accepted 31 December 2019 Available online 8 January 2020 Dedicated to Dr. Milan Práger on his 90th birthday Keywords: Weak solution Nonlinear heat conduction Heat transfer coefficient Hölder continuity
1. Introduction The problem of a stationary heat conduction in nonhomogeneous, anisotropic, and nonlinear media with mixed boundary conditions consists of finding u ∈ C 1 (Ω) ∩ C 2 (Ω) such that −div(A(x, u) grad u) = F (x, u) in
Ω,
(1)
on
Γ1 ,
(2)
n A(s, u) grad u = G(s, u) on
Γ2 ,
(3)
u =u ⊤
where Ω ⊂ Rd , d ∈ {1, 2, . . . }, is a bounded domain with Lipschitz continuous boundary ∂Ω, n is the outward unit normal to ∂Ω, Γ1 , Γ2 ⊂ ∂Ω are relatively open sets with respect to the topology on ∂Ω such that Γ1 ∩ Γ2 = ∅, Γ 1 ∪ Γ 2 = ∂Ω, u is the unknown temperature, and u ∈ W21 (Ω) is a given temperature. Further, we assume that A = (Aij )di,j=1 is a measurable matrix function of heat conductivities such that ess supx,ξ,i,j |Aij (x, ξ)| ≤ C. Suppose that there exist positive constants CE and CH such that for almost all x ∈ Ω we have η ⊤ A(x, ξ)η ≥ CE ∥η∥2 ∀ξ ∈ R1 ∀η ∈ Rd (4) E-mail address: [email protected]. https://doi.org/10.1016/j.aml.2020.106214 0893-9659/© 2020 Published by Elsevier Ltd.
2
M. Křížek / Applied Mathematics Letters 103 (2020) 106214
and α
|Aij (x, ζ) − Aij (x, ξ)| ≤ CH |ζ − ξ| ∀ζ, ξ ∈ R1 , i, j = 1, . . . , d, (5) [1 ] where ∥ · ∥ stands for the Euclidean norm and α ∈ 2 , 1 is a given H¨older exponent, i.e., A(·, ·) is α-H¨older continuous with respect to the last variable. Finally, F (·, v(·)) ∈ L2 (Ω) and G(·, v(·)) ∈ L2 (Γ2 ) for any v ∈ W21 (Ω), and let for almost all x ∈ Ω and almost all s ∈ Γ2 the functions v ∈ R1 ↦→ F (x, v) ∈ R1
and v ∈ R1 ↦→ G(s, v) ∈ R1
be nonincreasing.
(6)
In particular, these functions can be independent of v. The function F may have, for example, the following frequently used form F (x, v) = f (x) − c(x)v,
x ∈ Ω,
where f is the density of volume heat sources and c ≥ 0. Similarly, we can set G(s, v) = g(s) − h(s)v for s ∈ Γ2 , where g is the density of surface heat sources and h ≥ 0 is the heat transfer coefficient. Several real-life applications of the problem (1)–(3) are given in [1]. In [2] we proved the uniqueness of the classical solution of the above problem (1)–(3) only for a diagonal matrix A and for Lipschitz continuous heat conductivities Aij with respect to the last variable, Aij (x, ·) ∈ 1 (R) for almost all x ∈ Ω, i.e., when α = 1 in inequality (5). This technique was generalized in [3] to W∞ prove the uniqueness of the weak solution of (1)–(3). Note that we could not use the theory of monotone operators, since the operator corresponding to (1)–(3) is nonmonotone, see [3]. Moreover, this operator is nonpotential, which means that the problem (1)–(3) cannot be transformed to the minimization of some functional [1]. If an elliptic equation is not in the divergence form there exist examples of non-unique solutions, see [4, p. 209], [5]. Also semiconductor elliptic equations (describing e.g. thyristor) may have several different solutions for certain fixed boundary conditions, see [6, p. 43] and [7]. In [3, p. 182] we present a simple one-dimensional example of nonuniqueness of the weak solution of the stationary heat conduction problem (1)–(3), when A is continuous, but not Lipschitz continuous. This example shows that there exist infinitely many solutions for any kind of boundary conditions. In this paper we show that the uniqueness of the weak solution can be guaranteed also for 21 -H¨ older continuous heat conductivities. 2. Main uniqueness theorem Throughout this paper we shall use the standard Sobolev space notation, (·, ·)0,Ω and ⟨·, ·⟩0,Γ2 will denote the usual scalar product in (L2 (Ω))m , m = 1, 2, . . . , and L2 (Γ2 ), respectively. For simplicity, in what follows we shall often write only A(u) instead of A(x, u(x)), F (u) instead of F (x, u(x)), G(u) instead of G(s, u(s)), i.e., we shall not explicitly indicate the space variable x and the boundary variable s. Set V = {v ∈ W21 (Ω) | v = 0 on Γ1 }. Definition 1. A function u ∈ W21 (Ω) is called a weak solution of the problem (1)–(3) if u − u ∈ V, (A(u)grad u, grad v)0,Ω = (F (u), v)0,Ω + ⟨G(u), v⟩0,Γ2
(7) ∀v ∈ V.
(8)
Theorem 1. Let (4), (5), and (6) hold, let u1 and u2 be two weak solutions of the problem (7)–(8), and let Γ1 be nonempty. Then u1 = u2 a.e. in Ω.
M. Křížek / Applied Mathematics Letters 103 (2020) 106214
3
Fig. 1. Schematic illustration of function values of vε over Ω .
Proof . Define the following set Ω0 = {x ∈ Ω | u2 (x) > u1 (x)} and suppose that the d-dimensional Lebesgue measure of this set is positive, i.e. measd Ω0 > 0.
(9)
Next we will show that this inequality does not hold. Let ε > 0 be arbitrary and put (see Fig. 1) Ωε = {x ∈ Ω0 | u2 − u1 > ε}.
(10)
We shall use a special test function in (8), namely, { min(ε, u2 − u1 ) in Ω0 , vε = 0 in Ω \ Ω0 .
(11)
Using (7), we find that u2 −u, u1 −u ∈ V and thus u2 −u1 ∈ V . Consequently, the positive part (u2 −u1 )+ lies also in V . The reason is that v ↦→ v + represents a continuous mapping from W21 (Ω) into itself, see [8, p. 29] or [9, p. 58]. The mapping v ↦→ |v| = v + + v − is continuous, too. Hence, the relation (11) and the equality 1 min(a, b) = (a + b − |a − b|) 2 yield that vε = min(ε, (u2 − u1 )+ ) ∈ W21 (Ω ) and vε ∈ V, (12) since Γ1 ∩ Ω0 = ∅. According to (8) and (12), we find that (A(u1 )grad u1 , grad vε )0,Ω =(A(u2 )grad u2 , grad vε )0,Ω + (F (u1 ) − F (u2 ), vε )0,Ω + ⟨G(u1 ) − G(u2 ), vε ⟩0,Γ2 .
(13)
Clearly vε ≥ 0 in Ω. Thus from (6) we obtain (F (u1 ) − F (u2 ))vε ≥ 0
on Ω,
(G(u1 ) − G(u2 ))vε ≥ 0
From (4), (11), (12), (13), and (14) we get ∫ ∫ CE ∥grad vε ∥2 dx ≤ (grad vε )⊤ A(u1 )grad vε dx Ω ∫Ω = (grad vε )⊤ A(u1 )grad (u2 − u1 )dx Ω0 \Ωε
on Γ2 .
(14)
M. Křížek / Applied Mathematics Letters 103 (2020) 106214
4
∫ =
(grad vε )⊤ (A(u1 )grad u2 − A(u1 )grad u1 )dx
∫Ω
(grad vε )⊤ (A(u1 )grad u2 − A(u2 )grad u2 )dx ∫ − (F (u1 ) − F (u2 ))vε dx − (G(u1 ) − G(u2 ))vε ds Ω Γ2 ∫ ≤ (grad vε )⊤ (A(u1 ) − A(u2 ))grad u2 dx.
=
Ω∫
(15)
Ω
The last integral can be estimated by means of (10), (11), the Cauchy–Schwarz inequality, and (5) as follows ∫ (grad vε )⊤ (A(u1 ) − A(u2 ))grad u2 dx Ω0 \Ωε (∫ )1/2 (∫ )1/2 2 ≤ ∥(A(u1 ) − A(u2 ))grad u2 ∥ dx ∥grad vε ∥2 dx Ω0 \Ωε Ω0 \Ωε (∫ )1/2 (16) ≤ εα CH d1/2 ∥grad vε ∥0,Ω0 \Ωε . ∥grad u2 ∥2 dx Ω0 \Ωε
From this and (15) we find that there exists a constant C0 > 0 such that ∥grad vε ∥0,Ω ≤ C0 εα ∥grad u2 ∥0,Ω0 \Ωε .
(17)
According to the generalized Friedrichs inequality, there exists a constant C1 (p) > 0 such that ∥v∥0,p ≤ C1 (p)∥grad v∥0,p
for p ≥ 1,
(18)
where ∥ · ∥0,p stands for the standard Lebesgue Lp -norm. Further, we shall use the imbedding (L2 (Ω))d ⊂ (Lp (Ω))d for p ∈ [1, 2], i.e., there exists a constant C2 (p) such that ∥v∥0,p ≤ C2 (p)∥v∥0,2
∀v ∈ L2 (Ω).
(19)
Now from relations (10), (12), (18), (19), and (17) squared with p = 2α [1
] for α ∈ 2 , 1 , we obtain
≤
∫
∫
p |vε | dx = ε−p ∥vε ∥p0,p ≤ ε−p C1p (p)∥grad vε ∥p0,p Ωε Ω ε−p C1p (p)C2 (p)∥grad vε ∥p0 ≤ C0 C1p (p)C2 (p)∥grad u2 ∥p0,Ω0 \Ωε → 0 as
measd Ωε = ε−p
εp dx ≤ ε−p
ε → 0,
(20)
where u2 ∈ V is fixed, grad vε ∈ (L2 (Ω))d , and the constant C0 C1p (p)C2 (p) > 0 is independent of ε. However, this contradicts (9) and (10), because measd Ωε → measd Ω0
as ε → 0.
Therefore, measd Ω0 = 0 and u1 ≥ u2 a.e. in Ω. Interchanging the role of u1 and u2 , we get u2 ≥ u1 a.e. in Ω, and thus the uniqueness follows. □ Remark 1. When p ∈ (0, 1), then the symbol ∥ · ∥0,p stands for the so-called quasi-norm which satisfies all norm axioms except for the usual triangle inequality which is replaced by ∥v + w∥0,p ≤ C3 (p)(∥v∥0,p + ∥w∥0,p ) with some constant C3 (p) > 1. A natural question arises whether the Friedrichs inequality (18) can be extended to the case p ∈ (0, 1).
M. Křížek / Applied Mathematics Letters 103 (2020) 106214
5
Remark 2. In [3], we were not able to prove uniqueness of the Galerkin solutions of (1)–(3), since the corresponding test function defined like in (12) does not belong to finite element spaces, in general. However, the uniqueness was proved quite recently in [10] by a different approach. The nonlinear discrete problem is usually transformed to a sequence of linear problems by the method of successive approximations (the freezing coefficient method), see [1]. 3. Further uniqueness theorems In this section, we prove similar statements as in Theorem 1 under different assumptions. Now the set Γ1 can be empty. Theorem 2. Let (4), (5), and (6) hold, let u1 and u2 be two weak solutions of the problem (7)–(8), and let there exist a constant C > 0 and a nonempty relatively open subset Γ3 ⊂ Γ2 such that v2 − v1 ≤ C(G(s, v1 ) − G(s, v2 )) for almost all s ∈ Γ3 and all v1 , v2 ∈ R1 , v2 ≥ v1 . Then u1 = u2 a.e. in Ω. Proof . Using (12), we get 0 ≤ vε ≤ (u2 − u1 )+ = u2 − u1 ≤ C(G(u1 ) − G(u2 ))
on
Γ3 ∩ Ω 0 .
Hence, vε2 ≤ C(G(u1 ) − G(u2 ))vε
on
Γ3 ,
(21)
since (see Fig. 1) vε = 0
on
Γ3 \ Ω 0 .
(22)
Applying the following Friedrichs inequality, (21), (22), (13), (4), the Cauchy–Schwarz inequality, (5), and the inequality (17), which is independent of particular boundary conditions, we get ∥vε ∥20,Ω ≤C1 (∥vε ∥20,Γ3 + ∥grad vε ∥20,Ω ) (∫ ) ≤C2 (G(u1 ) − G(u2 ))vε ds + ∥grad vε ∥20,Ω Γ3 (∫ ) ≤C3 (G(u1 ) − G(u2 ))vε ds + CE ∥grad vε ∥20,Ω Γ3 (∫ ≤C3 (grad vε )⊤ (A(u1 )grad u1 − A(u2 )grad u2 )dx Ω ∫ ∫ ) − (F (u1 ) − F (u2 ))vε dx + (grad vε )⊤ A(u1 )grad (u2 − u1 )dx Ω ∫ Ω ⊤ ≤C3 (grad vε ) (A(u1 ) − A(u2 ))grad u2 dx Ω0 \Ωε (∫ )1/2 α 1/2 ≤C3 ε CH d ∥grad u2 ∥2 dx ∥grad vε ∥0,Ω0 \Ωε Ω0 \Ωε ∫ ≤C4 ε2α ∥grad u2 ∥2 dx. Ω0 \Ωε
From this and (19) we obtain for p = 2α similarly to (20) that ∫ ∫ p measd Ωε ≤ ε−p |vε | dx ≤ C2 (p)ε−p ∥vε ∥20,Ω ≤ C4 (p) Ω
∥grad u2 ∥2 dx → 0
Ω0 \Ωε
which again leads to a contradiction like at the end of the proof of Theorem 1. □
as ε → 0,
(23)
6
M. Křížek / Applied Mathematics Letters 103 (2020) 106214
Theorem 3. Let (4), (5), and (6) hold, let u1 and u2 be two weak solutions of the problem (7)–(8), and let ˜ ⊂ Ω such that there exist a constant C > 0 and a nonempty open subset Ω v2 − v1 ≤ C(F (x, v1 ) − F (x, v2 )) ˜ and all v1 , v2 ∈ R1 , v1 ≥ v2 . Then u1 = u2 a.e. in Ω. for almost all x ∈ Ω The proof is similar to that of Theorem 2. We only exchange the role of F and G and instead of the Friedrichs inequality (23) we will use the generalized Poincar´e inequality (see e.g. [9, p. 113]) ∥vε ∥20,Ω ≤ C(∥vε ∥20,Ω˜ + ∥grad vε ∥20,Ω ). Acknowledgments The author is indebted to Jan Brandts, Lawrence Somer, and Tom´aˇs Vejchodsk´ y for fruitful discussions. This paper was supported by grants 18-09628S and 20-01074S of the Grant Agency of the Czech Republic and RVO 67985840 of the Czech Republic. References [1] M. Kˇr´ıˇ zek, P. Neittaanm¨ aki, Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications, Kluwer Academic Publishers, Dordrecht, 1996. [2] I. Hlav´ aˇ cek, M. Kˇr´ıˇ zek, On a nonpotential and nonmonotone second order elliptic problem with mixed boundary conditions, Stability Appl. Anal. Contin. Media 3 (1993) 85–97. [3] I. Hlav´ aˇ cek, M. Kˇr´ıˇ zek, J. Mal´ y, On Galerkin approximations of a quasilinear nonpotential elliptic problem of a nonmonotone type, J. Math. Anal. Appl. 184 (1994) 168–189. [4] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1977. [5] N.G. Meyers, An example of non-uniqueness in the theory of quasi-linear elliptic equations of second order, Arch. Ration. Mech. Anal. 14 (1963) 177–179. [6] P.A. Markowich, The Stationary Semiconductor Device Equations, Springer-Verlag, Wien, 1986. [7] M.S. Mock, An example of nonuniqueness of stationary solutions in semiconductor device model, COMPEL 1 (1982) 165–174. [8] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, Berlin, 1977. [9] J. Neˇ cas, Direct Methods in the Theory of Elliptic Equations, in: Springer Monographs in Mathematics, Springer, Heidelberg, 2012. [10] S. Pollock, Y. Zhu, Uniqueness of discrete solutions of nonmonotone PDEs without a globally fine mesh condition, Numer. Math. 139 (2018) 845–865.