Analytic solutions for the approximated Kantorovich mass transfer problems by p -Laplacian approach
aa r X i v : . [ m a t h . O C ] S e p ANALYTIC SOLUTIONS FOR THE APPROXIMATEDKANTOROVICH MASS TRANSFER PROBLEMS BY p -LAPLACIANAPPROACH YANHUA WU XIAOJUN LU
1. Department of Sociology, School of Public Administration, Hohai University,211189, Nanjing, China2. Department of Mathematics & Jiangsu Key Laboratory of Engineering Mechanics,Southeast University, 210096, Nanjing, China
Abstract.
This manuscript discusses the approximation of a global maximizer ofthe Kantorovich mass transfer problem through the approach of p -Laplacian equa-tion. Using an approximation mechanism, the primal maximization problem can betransformed into a sequence of minimization problems. By applying the canonicalduality theory, one is able to derive a sequence of analytic solutions for the mini-mization problems. In the final analysis, the convergence of the sequence to a globalmaximizer of the primal Kantorovich problem will be demonstrated. Introduction
Complementary variational method has been applied in the study of finite deforma-tion by Hellinger since the beginning of the 20th century. During the last few years,considerable effort has been taken to find minimizers for non-convex strain energy func-tionals with a double-well potential. In this respect, Ericksen bar is a typical modelfor the research of elastic phase transitions. In [10], R. W. Ogden et al. treated theEricksen bar as a 1-D smooth compact manifold and discussed two classical loadingdevices, namely, hard device and soft device, by introducing a distributed axial bodyforce. By applying the canonical duality method, the authors characterized the localenergy extrema and the global energy minimizer for both hard device and soft device.This method proved to be very efficient in solving lots of open problems in the mechan-ical fields such as non-convex optimal design and control, nonlinear stability analysisof finite deformation, nonlinear elastic theory with residual strain, existence results forNash equilibrium points of non-cooperative games etc. Interested readers can refer to[9, 10, 16] for more details.This paper mainly addresses the Kantorovich problem in higher dimensions. LetΩ = B ( O , R ) and Ω ∗ = B ( O , R ) denote two open balls with centers O and O , Corresponding author: Xiaojun Lu, Department of Mathematics & Jiangsu Key Laboratory ofEngineering Mechanics, Southeast University, 210096, Nanjing, China.Keywords: Kantorovich mass transfer, p -Laplacian problem, canonical duality theory.Mathematics Subject Classification: 35J20, 35J60, 49K20, 80A20. . Wu and X. Lu Kantorovich mass transfer problemradii R and R in the Euclidean space R n , respectively, and we denote U := Ω ∪ Ω ∗ . Here we focus on the following two representative cases: • Ω = B ( O , R ), Ω ∗ = B ( O , R ), Ω ∩ Ω ∗ = ∅ ; • Ω = B ( O , R ), Ω ∗ = B ( O , R ), R = R .Let f + and f − be two nonnegative density functions in Ω and Ω ∗ , respectively, andsatisfy the normalized balance condition Z Ω f + dx = Z Ω ∗ f − dx = 1 . For convenience’s sake, let f := f + − f − . First, let the admissible set A be defined as A := n φ ∈ W , ∞ ( U ) ∩ C ( U ) (cid:12)(cid:12)(cid:12) k∇ φ k L ∞ ≤ , φ radially symmetric , φ = 0 on Ω ∩ Ω ∗ o , where W , ∞ ( U ) is a Sobolev spaces. The aim is to find an analytic global maximizer(so-called Kantorovich potential ) u ∈ A for the Kantorovich problem in the followingform,(1) ( P ) : K [ u ] = max w ∈ A n K [ w ] := Z U wf dx o . In this paper, we consider the Kantorovich problem through a p -Laplacian approachby introducing an approximation of the primal ( P ) [6],(2) ( P ( p ) ) : min w p ∈ A n I ( p ) [ w p ] := Z U (cid:16) H ( p ) ( ∇ w p ) − w p f (cid:17) dx o , where p > H ( p ) : R n → R + is defined as H ( p ) ( γ ) := | γ | p /p, and I ( p ) is called the potential energy functional . It’s evident that − lim p → + ∞ min w p ∈ A { I ( p ) [ w p ] } = max w ∈ A { K [ w ] } . Consequently, once a function ¯ u p satisfying I ( p ) [¯ u p ] = min w p ∈ A { I ( p ) [ w p ] } is obtained, thenit will help find out an analytic Kantorovich potential u = lim p → + ∞ ¯ u p in the L ∞ sense,which maximizes the primal problem ( P ).By variational calculus, one derives a corresponding Euler-Lagrange equation for( P ( p ) ),(3) div( |∇ u p | p − ∇ u p ) + f = 0 , in U \ { Ω ∩ Ω ∗ } , equipped with the Dirichlet boundary condition. For the integer case, p = 1, byvariational calculus, one obtains the mean curvature operator; p = 2, one has theLaplace operator(see [8]). For p = n , one derives the n -harmonic equation which isinvariant under M¨obius transformation. While for the fractional case, such as p = 3 / p − Laplacian describes the flow through porous media. And glaciologist usually studythe case p ∈ (1 , / p -Laplacian problem which is difficult to solve by thedirect approach [3, 8, 15]. However, by the canonical duality theory, one is able to2. Wu and X. Lu Kantorovich mass transfer problemdemonstrate the existence and uniqueness of the solution for the nonlinear differentialequation, which establishes the equivalence between the global minimizer of ( P ( p ) ) andthe solution of Euler-Lagrange equation (3).At the moment, we would like to introduce the main theorems. Theorem 1.1.
For any positive density functions f + ∈ C (Ω) and f − ∈ C (Ω ∗ ) sat-isfying the normalized balance condition, there exists a unique solution ¯ u p ∈ A forthe Euler-Lagrange equation (3), which is at the same time a global minimizer for theapproximation problem ( P ( p ) ). In particular, let E p ( x ) := x (2 p − / ( p − , x ∈ [0 , , and E − p stands for the inverse of E p , then one has • Ω = B ( O , R ) , Ω ∗ = B ( O , R ) , Ω ∩ Ω ∗ = ∅ . ¯ u p can be represented explicitly as ¯ u p ( r ) = Z rR F ( ρ ) ρ/E − p ( F ( ρ ) ρ ) dρ, r ∈ [0 , R ] , Z rR G ( ρ ) ρ/E − p ( G ( ρ ) ρ ) dρ, r ∈ [0 , R ] , where F and G are defined as F ( r ) := − Γ( n/ / (2 π n/ r n ) + Z R r f + ( ρ ) ρ n − /r n dρ, r ∈ [0 , R ] ,G ( r ) := Γ( n/ / (2 π n/ r n ) − Z R r f − ( ρ ) ρ n − /r n dρ, r ∈ [0 , R ] . • Ω = B ( O , R ) , Ω ∗ = B ( O , R ) , R > R > . ¯ u p can be represented explicitlyas ¯ u p ( r ) = Z rR F p ( ρ ) ρ/E − p ( F p ( ρ ) ρ ) dρ, r ∈ [ R , R ] , where F p ( r ) := C p R n /r n − Z rR f + ( ρ ) ρ n − /r n dρ, and C p ∈ (0 , Γ( n/ / (2 π n/ R n )) . • Ω = B ( O , R ) , Ω ∗ = B ( O , R ) , R > R > . ¯ u p can be represented explicitlyas ¯ u p ( r ) = Z rR G p ( ρ ) ρ/E − p ( G p ( ρ ) ρ ) dρ, r ∈ [ R , R ] , where G p ( r ) := − D p R n /r n + Z rR f − ( ρ ) ρ n − /r n dρ, and D p ∈ (0 , Γ( n/ / (2 π n/ R n )) .
3. Wu and X. Lu Kantorovich mass transfer problem
Theorem 1.2.
For any positive density functions f + ∈ C (Ω) and f − ∈ C (Ω ∗ ) sat-isfying the normalized balance condition, there exists a global maximizer for the Kan-torovich problem ( P ). The rest of the paper is organized as follows. In Section 2, first we introduce someuseful notations which will simplify our proof considerably. Then we apply the canoni-cal dual transformation to deduce a perfect dual problem ( P ( p ) d ) corresponding to ( P ( p ) )and a pure complementary energy principle. Next we apply the canonical duality the-ory to prove Theorem 1.1 and Theorem 1.2.2. Proof of the main results
Some useful notations. • −→ θ p is given by −→ θ p ( x ) = ( θ p, ( x ) , · · · , θ p,n ( x )) = |∇ w p | p − ∇ w p . • Φ ( p ) : A → L ∞ ( U ) is a nonlinear geometric mapping defined asΦ ( p ) ( w p ) := |∇ w p | . For convenience’s sake, denote ξ p := Φ ( p ) ( w p ) . It is evident that ξ p belongs tothe function space U given by U := n φ ∈ L ∞ ( U ) (cid:12)(cid:12)(cid:12) ≤ φ ≤ o . • Ψ ( p ) : U → L ∞ ( U ) is a canonical energy defined asΨ ( p ) ( ξ p ) := ξ p/ p /p, which is a convex function with respect to ξ p . For simplicity, denote ζ p := ξ ( p − / p /
2, which is the Gˆateaux derivative of Ψ ( p ) with respect to ξ p . Moreover, ζ p is invertible with respect to ξ p and belongs to the function space W , W := n φ ∈ L ∞ ( U ) (cid:12)(cid:12)(cid:12) ≤ φ ≤ / o . • Ψ ( p ) ∗ : W → L ∞ ( U ) is defined asΨ ( p ) ∗ ( ζ p ) := ξ p ζ p − Ψ ( p ) ( ξ p ) = (1 − /p )2 / ( p − ζ p/ ( p − p . • λ p is defined as λ p := 2 ζ p , and belongs to the function space V , V := n φ ∈ L ∞ ( U ) (cid:12)(cid:12)(cid:12) ≤ φ ≤ o . Canonical duality techniques.Definition 2.1.
By Legendre transformation, one defines a Gao-Strang total comple-mentary energy functional Ξ ( p ) , Ξ ( p ) ( u p , ζ p ) := Z U n Φ ( p ) ( u p ) ζ p − Ψ ( p ) ∗ ( ζ p ) − f u p o dx. Next we introduce an important criticality criterium for the Gao-Strang total com-plementary energy functional. 4. Wu and X. Lu Kantorovich mass transfer problem
Definition 2.2. (¯ u p , ¯ ζ p ) ∈ A × W is called a critical pair of Ξ ( p ) if and only if (4) D u p Ξ ( p ) (¯ u p , ¯ ζ p ) = 0 , (5) D ζ p Ξ ( p ) (¯ u p , ¯ ζ p ) = 0 , where D u p , D ζ p denote the partial Gˆateaux derivatives of Ξ ( p ) , respectively. Indeed, by variational calculus, we have the following observation from (4) and (5).
Lemma 2.3.
On the one hand, for any fixed ζ p ∈ W , (3 . is equivalent to the equi-librium equation div( λ p ∇ ¯ u p ) + f = 0 , in U \ { Ω ∩ Ω ∗ } . On the other hand, for any fixed u p ∈ A , (5) is consistent with the constructive law Φ ( p ) ( u p ) = D ζ p Ψ ( p ) ∗ ( ¯ ζ p ) . Lemma 3.2.3 indicates that ¯ u p from the critical pair (¯ u p , ¯ ζ p ) solves the Euler-Lagrangeequation (3). Definition 2.4.
From Definition 3.2.1, one defines the Gao-Strang pure complemen-tary energy I ( p ) d in the form I ( p ) d [ ζ p ] := Ξ ( p ) (¯ u p , ζ p ) , where ¯ u p solves the Euler-Lagrange equation (3). To simplify the discussion, we use another representation of the pure energy I ( p ) d given by the following lemma. Lemma 2.5.
The pure complementary energy functional I ( p ) d can be rewritten as I ( p ) d [ ζ p ] = − Z U n |−→ θ p | / (4 ζ p ) + (1 − /p )2 / ( p − ζ p/ ( p − p o , where −→ θ p satisfies (6) div −→ θ p + f = 0 in U, equipped with a hidden boundary condition.Proof. Through integrating by parts, one has I ( p ) d [ ζ p ] = − Z U n div(2 ζ p ∇ ¯ u p ) + f o ¯ u p dx | {z } ( I ) − Z U n ζ p |∇ ¯ u p | + (1 − /p )2 / ( p − ζ p/ ( p − p o dx. | {z } ( II ) Since ¯ u p solves the Euler-Lagrange equation (3), then the first part ( I ) disappears.Keeping in mind the definition of −→ θ p and ζ p , one reaches the conclusion. (cid:3)
5. Wu and X. Lu Kantorovich mass transfer problemWith the above discussion, next we establish a variational problem to the approxi-mation problem ( P ( p ) ).(7) ( P ( p ) d ) : max ζ p ∈ W n I ( p ) d [ ζ p ] = − Z U n |−→ θ p | / (4 ζ p ) + (1 − /p )2 / ( p − ζ p/ ( p − p o . Indeed, by calculating the Gˆateaux derivative of I ( p ) d with respect to ζ p , one has Lemma 2.6.
The variation of I ( p ) d with respect to ζ p leads to the dual algebraic equation(DAE), namely, (8) |−→ θ p | = (2 ¯ ζ p ) (2 p − / ( p − , where ¯ ζ p is from the critical pair (¯ u p , ¯ ζ p ) . Taking into account the notation of λ p , the identity (8) can be rewritten as(9) |−→ θ p | = E p ( λ p ) = λ (2 p − / ( p − p . It is evident E p is monotonously increasing with respect to λ ∈ [0 , Proof of Theorem 1.1.
From the above discussion, one deduces that, once θ p isgiven, then the analytic solution of the Euler-Lagrange equation (3) can be representedas(10) ¯ u p ( x ) = Z xx η p ( t ) dt, where x ∈ U , x ∈ ∂U , η p := θ p /λ p . Together with (9), one sees that lim p → + ∞ |∇ ¯ u p | = 1 , which is consistent with the a-priori estimate in [5]. Next we verify that ¯ u p is exactlya global minimizer for ( P ( p ) ) and ¯ ζ p is a global maximizer for ( P ( p ) d ). Lemma 2.7. (Canonical duality theory) For any positive density functions f + ∈ C (Ω) and f − ∈ C (Ω ∗ ) satisfying the normalized balance condition, there exists a uniqueradially symmetric solution ¯ u p ∈ A for the Euler-Lagrange equations (3) with Dirichletboundary in the form of (10), which is a unique global minimizer over A for theapproximation problem ( P ( p ) ). And the corresponding ¯ ζ p is a unique global maximizerover W for the dual problem ( P ( p ) d ). Moreover, the following duality identity holds, (11) I ( p ) (¯ u p ) = min u p ∈ A I ( p ) ( u p ) = Ξ ( p ) (¯ u p , ¯ ζ p ) = max ζ p ∈ W I ( p ) d ( ζ p ) = I ( p ) d ( ¯ ζ p ) . Lemma 3.2.7 shows that the maximization of the pure complementary energy func-tional I ( p ) d is perfectly dual to the minimization of the potential energy functional I ( p ) .Indeed, identity (11) indicates there is no duality gap between them. Proof.
We divide our proof into three parts. In the first and second parts, we discuss theuniqueness of θ p for both cases. Global extremum will be studied in the third part. Itis worth noticing that the first and second parts are similar to the proof of Theorem 1.2. First Part:
Ω = B ( O , R ) , Ω ∗ = B ( O , R ) , Ω ∩ Ω ∗ = ∅
6. Wu and X. Lu Kantorovich mass transfer problem (1) Discussion in
ΩLet O = ( a , a , · · · , a n ). Actually, a radially symmetric solution for the Euler-Lagrange equation (3) is of the form −→ θ p = F p ( r )(( x − a , · · · , x n − a n )) = F p (cid:16)vuut n X i =1 ( x i − a i ) (cid:17) (( x − a , · · · , x n − a n )) , where F p ( r ) = C p R n /r n + Z R r f + ( ρ ) ρ n − /r n dρ is the unique solution of the differential equation F ′ p ( r ) + nF p ( r ) /r = − f + ( r ) /r, r ∈ (0 , R ] . Recall that ¯ u p ( R ) = 0, as a result,¯ u p ( r ) = Z rR (cid:16) R n C p + Z R ρ f + ( r ) r n − dr (cid:17) / (cid:16) ρ n − λ p ( ρ ) (cid:17) dρ, r ∈ (0 , R ] . As a matter of fact, if ¯ u p ∈ C [0 , R ], we havelim ρ → + n R n C p + Z R ρ f + ( r ) r n − dr o = 0 , which indicates C p = − Γ( n/ / (2 π n/ R n ) , from the normalized balance condition Z Ω f + ( x ) dx = 2 π n/ / Γ( n/ Z R f + ( r ) r n − dr = 1 . (2) Discussion in Ω ∗ Let O = ( b , b , · · · , b n ). In fact, a radially symmetric solution for the Euler-Lagrange equation (3) is of the form −→ θ p = G p ( r )(( x − b , · · · , x n − b n )) = G p (cid:16)vuut n X i =1 ( x i − b i ) (cid:17) (( x − b , · · · , x n − b n )) , where G p ( r ) = D p R n /r n − Z R r f − ( ρ ) ρ n − /r n dρ is the unique solution of the differential equation G ′ p ( r ) + nG p ( r ) /r = f − ( r ) /r, r ∈ (0 , R ] . Recall that ¯ u p ( R ) = 0, as a result,¯ u p ( r ) = Z rR (cid:16) R n D p − Z R ρ f − ( r ) r n − dr (cid:17) / (cid:16) ρ n − λ p ( ρ ) (cid:17) dρ, r ∈ (0 , R ] .
7. Wu and X. Lu Kantorovich mass transfer problemIndeed, if ¯ u p ∈ C [0 , R ], then by applying the similar contradiction method as above,one has lim ρ → + n R n D p − Z R ρ f − ( r ) r n − dr o = 0 , which indicates D p = Γ( n/ / (2 π n/ R n )from the normalized balance condition Z Ω ∗ f − ( x ) dx = 2 π n/ / Γ( n/ Z R f − ( r ) r n − dr = 1 . Second Part:
Ω = B ( O , R ) , Ω ∗ = B ( O , R ) , R = R (1) R > R > O = ( a , a , · · · , a n ). Actually, a radially symmetric solution for the Euler-Lagrange equation (3) is of the form −→ θ p = F p ( r )(( x − a , · · · , x n − a n )) = F p (cid:16)vuut n X i =1 ( x i − a i ) (cid:17) (( x − a , · · · , x n − a n )) , where F p ( r ) = C p R n /r n − Z rR f + ( ρ ) ρ n − /r n dρ is the unique solution of the differential equation F ′ p ( r ) + nF p ( r ) /r = − f + ( r ) /r, r ∈ [ R , R ] . Recall that ¯ u p ( R ) = 0, consequently,¯ u p ( r ) = Z rR (cid:16) R n C p − Z ρR f + ( r ) r n − dr (cid:17) / (cid:16) ρ n − λ p ( ρ ) (cid:17) dρ, r ∈ [ R , R ] . Let ˜ F ( r ) := 1 /R n Z rR f + ( ρ ) ρ n − dρ, r ∈ [ R , R ] . Since f + >
0, then ˜ F ∈ C [ R , R ] is a strictly increasing function with respect to r ∈ [ R , R ] and consequently is invertible. Let ˜ F − be its inverse function, which isalso a strictly increasing function. From (9), we see that there exists a unique piecewisecontinuous function λ p ( x ) ≥
0. Sincelim r → ˜ F − ( C p ) ( − ˜ F ( r ) + C p ) R n / ( r n − λ p ( r )) = 0 , thus ¯ u p is continuous at the point r = ˜ F − ( C p ). As a result, ¯ u p ∈ C [ R , R ]. Noticethat ¯ u p ( R ) = 0 and we can determine the constant C p uniquely. Indeed, let µ p ( ρ, t ) := (cid:16) R n t − Z ρR f + ( r ) r n − dr (cid:17) / (cid:16) ρ n − λ p ( ρ, t ) (cid:17)
8. Wu and X. Lu Kantorovich mass transfer problemand M k ( t ) := Z ˜ F − ( t ) R µ p ( ρ, t ) dρ + Z R ˜ F − ( t ) µ p ( ρ, t ) dρ, where λ p ( ρ, t ) is from (9). It is evident that λ p depends on C p . As a matter of fact, itis easy to check M is strictly increasing with respect to t , which leads to C p = M − p (0) . Furthermore, by a similar discussion as in [16], we havelim k →∞ C p = ˜ F (( R + R ) / . (2) < R < R In fact, a radially symmetric solution for the Euler-Lagrange equation (3) is of theform −→ θ p = G p ( r )(( x − a , · · · , x n − a n )) = G p (cid:16)vuut n X i =1 ( x i − a i ) (cid:17) (( x − a , · · · , x n − a n )) , where G p ( r ) = − D p R n /r n + Z rR f − ( ρ ) ρ n − /r n dρ is the unique solution of the differential equation G ′ p ( r ) + nG p ( r ) /r = f − ( r ) /r, r ∈ [ R , R ] . Recall that ¯ u p ( R ) = 0, as a result,¯ u p ( r ) = Z rR (cid:16) − R n D p + Z ρR f − ( r ) r n − dr (cid:17) / (cid:16) ρ n − λ p ( ρ ) (cid:17) dρ, r ∈ [ R , R ] . Let ˜ G ( r ) := 1 /R n Z rR f − ( ρ ) ρ n − dρ, r ∈ [ R , R ] . Since f − >
0, then ˜ G ∈ C [ R , R ] is a strictly increasing function with respect to r ∈ [ R , R ] and consequently is invertible. Let ˜ G − be its inverse function, which isalso a strictly increasing function. From (9), we see that there exists a unique piecewisecontinuous function λ p ( x ) ≥
0. Sincelim r → G − ( D p ) ( G ( r ) − D p ) R n / ( r n − λ p ( r )) = 0 , thus ¯ u p is continuous at the point r = G − ( D p ). As a result, ¯ u p ∈ C [ R , R ]. Noticethat ¯ u p ( R ) = 0 and we can determine the constant D p uniquely. Indeed, let η p ( ρ, t ) := (cid:16) − R n t + Z ρR f − ( r ) r n − dr (cid:17) / (cid:16) ρ n − λ p ( ρ, t ) (cid:17) and N p ( t ) := Z ˜ G − ( t ) R η p ( ρ, t ) dρ + Z R ˜ G − ( t ) η p ( ρ, t ) dρ,
9. Wu and X. Lu Kantorovich mass transfer problemwhere λ p ( ρ, t ) is from (9). It is evident that λ p depends on C p . As a matter of fact, itis easy to check N p is strictly increasing with respect to t , which leads to D p = N − p (0) . Furthermore, by a similar discussion as in [16], we havelim k →∞ D p = ˜ G (( R + R ) / . Third Part:
On the one hand, for any test function φ ∈ A satisfying ∇ φ = 0 a.e. in U , thesecond variational form δ φ I ( p ) with respect to φ is equal to(12) Z U n |∇ ¯ u p | p − |∇ φ | + ( p − |∇ ¯ u p | p − ( ∇ ¯ u p · ∇ φ ) o dx. On the other hand, for any test function ψ ∈ W satisfying ψ = 0 a.e. in U , the secondvariational form δ ψ I ( p ) d with respect to ψ is equal to(13) − Z U ψ n |−→ θ p | / (2 ¯ ζ p ) + 1 / ( p − p/ ( p − ζ (4 − p ) / ( p − p o dx. From (12) and (13), one deduces immediately that δ φ I ( p ) (¯ u p ) > , δ ψ J ( p ) d ( ¯ ζ p ) < . Together with the uniqueness of −→ θ p discussed in the first and second parts, the proofis concluded. (cid:3) Consequently, we reach the conclusion of Theorem 1.1 by summarizing the abovediscussion.2.4.
Proof of Theorem 1.2:
According to Rellich-Kondrachov Compactness Theo-rem, since sup k | ¯ u k | ≤ C ( R , R )and sup k |∇ ¯ u k | ≤ , then, there exists a subsequence(without any confusion, we still denote as) { ¯ u k } k and u ∈ W , ∞ ( U ) ∩ C ( U ) such that(14) ¯ u k → u ( k → ∞ ) in L ∞ ( U ) , (15) ∇ ¯ u k ∗− ⇀ ∇ u ( k → ∞ ) weakly ∗ in L ∞ ( U ) . It remains to check that u satisfies (5). From (19), one has k∇ u k L ∞ ( U ) ≤ lim inf k →∞ k∇ ¯ u k k L ∞ ( U ) ≤ sup k →∞ k∇ ¯ u k k L ∞ ( U ) ≤ . Consequently, one reaches the conclusion of Theorem 1.2 by summarizing the abovediscussion. 10. Wu and X. Lu Kantorovich mass transfer problem
Remark 2.8.
Frankly speaking, the uniqueness of the global minimizer for the primalproblem does not hold when U is a general Lipschitz domain. As p → ∞ , we have theinfinity harmonic equation n X j,k =1 ∂u∂x j ∂u∂x k ∂ u∂x j ∂x k = f. This equation has often been used in image processing and optimal Lipschitz extensions.
Acknowledgment : This project is partially supported by US Air Force Office ofScientific Research (AFOSR FA9550-10-1-0487), Natural Science Foundation of JiangsuProvince (BK 20130598), National Natural Science Foundation of China (NSFC 71673043,71273048, 71473036, 11471072), the Scientific Research Foundation for the ReturnedOverseas Chinese Scholars, Fundamental Research Funds for the Central Universitieson the Field Research of Commercialization of Marriage between China and Vietnam(No. 2014B15214). This work is also supported by Open Research Fund Program ofJiangsu Key Laboratory of Engineering Mechanics, Southeast University (LEM16B06).
References [1] L. Ambrosio, Optimal transport maps in Monge-Kantorovich problem, ICM, 3(2002), 1-3.[2] L. Ambrosio, Lecture Notes on Optimal Transfer Problems, preprint.[3] J. Bourgain, H. Brezis, Sur l’´equation div u = f , C. R. Acad. Sci. Paris, Ser. I334(2002), 973-976.[4] L. A. Caffarelli, M. Feldman and R. J. MCcann, Constructing optimal maps for Monge’s transportproblem as a limit of strictly convex costs, Journal of AMS, 15(2001), 1-26.[5] L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer(survey paper).[6] L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich masstransfer problem, Mem. Amer. Math. Soc. 653(1999).[7] L. C. Evans, Three singular variational problems, preprint, 2002.[8] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, 2002.[9] D. Y. Gao, G. Strang, Geometric nonlinearity: Potential energy, complementary energy, and thegap function, Quart. Appl. Math. 47(3)(1989), 487-504.[10] D. Y. Gao, R. W. Ogden, Multiple solutions to non-convex variational problems with implicationsfor phase transitions and numerical computation. Q. Jl Mech. Appl. Math. , 497-522(2008)[11] D. Y. Gao and X. Lu, Z. Angew. Math. Phys. (2016), DOI 10.1007/s00033-016-0636-0.[12] W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math., 177(1996),113-161.[13] L. V. Kantorovich, On the transfer of masses, Dokl. Akad. Nauk. SSSR 37(1942), 227-229 (Rus-sian).[14] L. V. Kantorovich, On a problem of Monge, Uspekhi Mat. Nauk. 3(1948), 225-226.[15] J. L. Lions and E. Magenes, Probl`emes aux limites non homog`enes et applications I-III, Dunod,Paris, 1968-1970.[16] X. Lu and D. Y. Gao, Analytic solution for the 1-D Kantorovich mass transfer problem, preprint,2015.[17] Y. Wu and X. Lu, An approximation method for the optimization of p -th moment of R n -valuedrandom variable. preprint, 2015.[18] G. Monge, M´emoire sur la th´eorie des d´eblais et de remblais, Histoire de l’Acad´emie Royale desSciences de Paris, avec les M´emoire de Math´ematique et de Physique pour la mˆeme ann´ee, (1781),666-704.[19] S. T. Rachev: The Monge-Kantorovich mass transferance problem and its stochastic applications.Theory of Prob. and Appl., , 647-676(1984)
11. Wu and X. Lu Kantorovich mass transfer problem [20] V. N. Sudakov, Geometric problems in the theory of infinite-dimensional probability distributions,Proceedings of Steklov Institute 141(1979), 1-178.[21] A. M. Vershik, Some remarks on the infinite-dimensional problems of linear programming, RussianMath. Survey, 25(1970), 117-124.[20] V. N. Sudakov, Geometric problems in the theory of infinite-dimensional probability distributions,Proceedings of Steklov Institute 141(1979), 1-178.[21] A. M. Vershik, Some remarks on the infinite-dimensional problems of linear programming, RussianMath. Survey, 25(1970), 117-124.