The Gagliardo-Nirenberg inequality in bounded domains In many problems coming from the theory of partial differential equations, one has to deal with functions whose domain is not the whole Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , but rather some given bounded, open and connected set Ω ⊂ R n . {\displaystyle \Omega \subset \mathbb {R} ^{n}.} In the following, we also assume that Ω {\displaystyle \Omega } has finite Lebesgue measure and satisfies the cone condition (among those are the widely used Lipschitz domains ). Both Gagliardo and Nirenberg found out that their theorem could be extended to this case adding a penalization term to the right hand side. Precisely,
Theorem [17] (Gagliardo-Nirenberg in bounded domains) — Let Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} be a measurable, bounded, open and connected domain satisfying the cone condition. Let 1 ≤ q ≤ + ∞ {\displaystyle 1\leq q\leq +\infty } be a positive extended real quantity. Let j {\displaystyle j} and m {\displaystyle m} be non-negative integers such that j < m {\displaystyle j<m} . Furthermore, let 1 ≤ r ≤ + ∞ {\displaystyle 1\leq r\leq +\infty } be a positive extended real quantity, p ≥ 1 {\displaystyle p\geq 1} be real and θ ∈ [ 0 , 1 ] {\displaystyle \theta \in [0,1]} such that the relations 1 p = j n + θ ( 1 r − m n ) + 1 − θ q , j m ≤ θ ≤ 1 {\displaystyle {\dfrac {1}{p}}={\dfrac {j}{n}}+\theta \left({\dfrac {1}{r}}-{\dfrac {m}{n}}\right)+{\dfrac {1-\theta }{q}},\qquad {\dfrac {j}{m}}\leq \theta \leq 1} hold. Then, ‖ D j u ‖ L p ( Ω ) ≤ C ‖ D m u ‖ L r ( Ω ) θ ‖ u ‖ L q ( Ω ) 1 − θ + C ‖ u ‖ L σ ( Ω ) {\displaystyle \|D^{j}u\|_{L^{p}(\Omega )}\leq C\|D^{m}u\|_{L^{r}(\Omega )}^{\theta }\|u\|_{L^{q}(\Omega )}^{1-\theta }+C\|u\|_{L^{\sigma }(\Omega )}} where u ∈ L q ( Ω ) {\displaystyle u\in L^{q}(\Omega )} such that D m u ∈ L r ( Ω ) {\displaystyle D^{m}u\in L^{r}(\Omega )} and σ {\displaystyle \sigma } is arbitrary, with one exceptional case:
if r > 1 {\displaystyle r>1} and m − j − n r {\displaystyle m-j-{\frac {n}{r}}} is a non-negative integer, then the additional assumption j m ≤ θ < 1 {\displaystyle {\frac {j}{m}}\leq \theta <1} (notice the strict inequality) is needed. In any case, the constant C > 0 {\displaystyle C>0} depends on the parameters j , m , n , q , r , θ {\displaystyle j,\,m,\,n,\,q,\,r,\,\theta } , on the domain Ω {\displaystyle \Omega } , but not on u {\displaystyle u} .
The necessity of a different formulation with respect to the case Ω = R n {\displaystyle \Omega =\mathbb {R} ^{n}} is rather straightforward to prove. Indeed, since Ω {\displaystyle \Omega } has finite Lebesgue measure, any affine function belongs to L p ( Ω ) {\displaystyle L^{p}(\Omega )} for every p {\displaystyle p} (including p = + ∞ {\displaystyle p=+\infty } ). Of course, it holds much more: affine functions belong to C ∞ ( Ω ) {\displaystyle C^{\infty }(\Omega )} and all their derivatives of order greater than or equal to two are identically equal to zero in Ω {\displaystyle \Omega } . It can be easily seen that the Gagliardo-Nirenberg inequality for the case Ω = R n {\displaystyle \Omega =\mathbb {R} ^{n}} fails to be true for any non constant affine function, since a contradiction is immediately achieved when j = 1 {\displaystyle j=1} and m ≥ 2 {\displaystyle m\geq 2} , and therefore cannot hold in general for integrable functions defined on bounded domains.
That being said, under slightly stronger assumptions, it is possible to recast the theorem in such a way that the penalization term is "absorbed" in the first term at right hand side. Indeed, if u ∈ L q ( Ω ) ∩ W m , r ( Ω ) {\displaystyle u\in L^{q}(\Omega )\cap W^{m,r}(\Omega )} , then one can choose σ = min ( r , q ) {\displaystyle \sigma =\min(r,q)} and get ‖ D j u ‖ L p ( Ω ) ≤ C ‖ D m u ‖ L r ( Ω ) θ ‖ u ‖ L q ( Ω ) 1 − θ + C ‖ u ‖ L min ( r , q ) ( Ω ) θ ‖ u ‖ L min ( r , q ) ( Ω ) 1 − θ ≤ C ‖ u ‖ W m , r ( Ω ) θ ‖ u ‖ L q ( Ω ) 1 − θ . {\displaystyle {\begin{aligned}\|D^{j}u\|_{L^{p}(\Omega )}&\leq C\|D^{m}u\|_{L^{r}(\Omega )}^{\theta }\|u\|_{L^{q}(\Omega )}^{1-\theta }+C\|u\|_{L^{\min(r,q)}(\Omega )}^{\theta }\|u\|_{L^{\min(r,q)}(\Omega )}^{1-\theta }\\&\leq C\|u\|_{W^{m,r}(\Omega )}^{\theta }\|u\|_{L^{q}(\Omega )}^{1-\theta }.\end{aligned}}} This formulation has the advantage of recovering the structure of the theorem in the full Euclidean space, with the only caution that the Sobolev seminorm is replaced by the full W m , r {\displaystyle W^{m,r}} -norm. For this reason, the Gagliardo-Nirenberg inequality in bounded domains is commonly stated in this way. [18]
Finally, observe that the first exceptional case appearing in the statement of the Gagliardo-Nirenberg inequality for the whole space is no longer relevant in bounded domains, since for finite measure sets we have that L ∞ ( Ω ) ↪ L ρ ( Ω ) {\displaystyle L^{\infty }(\Omega )\hookrightarrow L^{\rho }(\Omega )} for any finite ρ ≥ 1. {\displaystyle \rho \geq 1.}
Generalization to non-integer orders The problem of interpolating different Sobolev spaces has been solved in full generality by Haïm Brezis and Petru Mironescu in two works dated 2018 and 2019. [10] [11] Furthermore, their results do not depend on the dimension n {\displaystyle n} and allow real values of j {\displaystyle j} and m {\displaystyle m} , rather than integer. Here, Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} is either the full space, a half-space or a bounded and Lipschitz domain. If s ∈ ( 0 , 1 ) {\displaystyle s\in (0,1)} and p ≥ 1 {\displaystyle p\geq 1} is an extended real quantity, the space W s , p ( Ω ) {\displaystyle W^{s,p}(\Omega )} is defined as follows W s , p ( Ω ) := { { u ∈ L p ( Ω ) : | u ( x ) − u ( y ) | | x − y | s + n p ∈ L p ( Ω × Ω ) } if p < + ∞ , { u ∈ L ∞ ( Ω ) : | u ( x ) − u ( y ) | | x − y | s ∈ L ∞ ( Ω × Ω ) } if p = + ∞ ; ‖ u ‖ W s , p ( Ω ) := { ( ‖ u ‖ L p ( Ω ) p + ∫ Ω ∫ Ω | u ( x ) − u ( y ) | p | x − y | n + s p ) 1 p if p < + ∞ , ‖ u ‖ L ∞ ( Ω ) + ‖ u ( x ) − u ( y ) ( x − y ) s ‖ L ∞ ( Ω × Ω ) if p = + ∞ ; {\displaystyle W^{s,p}(\Omega ):={\begin{cases}\left\{u\in L^{p}(\Omega ):{\dfrac {|u(x)-u(y)|}{|x-y|^{s+{\frac {n}{p}}}}}\in L^{p}(\Omega \times \Omega )\right\}&\quad {\text{if }}p<+\infty ,\\\left\{u\in L^{\infty }(\Omega ):{\dfrac {|u(x)-u(y)|}{|x-y|^{s}}}\in L^{\infty }(\Omega \times \Omega )\right\}&\quad {\text{if }}p=+\infty ;\\\end{cases}}\qquad \|u\|_{W^{s,p}(\Omega )}:={\begin{cases}\left(\|u\|_{L^{p}(\Omega )}^{p}+\displaystyle \int _{\Omega }\int _{\Omega }{\dfrac {|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}}\right)^{\frac {1}{p}}&\quad {\text{if }}p<+\infty ,\\\|u\|_{L^{\infty }(\Omega )}+\left\|{\dfrac {u(x)-u(y)}{(x-y)^{s}}}\right\|_{L^{\infty }(\Omega \times \Omega )}&\quad {\text{if }}p=+\infty ;\end{cases}}} and if s ≥ 1 {\displaystyle s\geq 1} we set W s , p ( Ω ) := { u ∈ W ⌊ s ⌋ , p ( Ω ) : D ⌊ s ⌋ u ∈ W { s } , p ( Ω ) } , ‖ u ‖ W s , p ( Ω ) := { ( ‖ u ‖ L p ( Ω ) p + ‖ D ⌊ s ⌋ u ‖ W { s } , p ( Ω ) p ) 1 p if p < + ∞ , ‖ u ‖ L ∞ ( Ω ) + ‖ D ⌊ s ⌋ u ‖ W { s } , ∞ ( Ω ) if p = + ∞ ; {\displaystyle W^{s,p}(\Omega ):=\{u\in W^{\lfloor s\rfloor ,p}(\Omega ):D^{\lfloor s\rfloor }u\in W^{\{s\},p}(\Omega )\},\qquad \|u\|_{W^{s,p}(\Omega )}:={\begin{cases}\left(\|u\|_{L^{p}(\Omega )}^{p}+\|D^{\lfloor s\rfloor }u\|_{W^{\{s\},p}(\Omega )}^{p}\right)^{\frac {1}{p}}&\quad {\text{if }}p<+\infty ,\\\|u\|_{L^{\infty }(\Omega )}+\|D^{\lfloor s\rfloor }u\|_{W^{\{s\},\infty }(\Omega )}&\quad {\text{if }}p=+\infty ;\end{cases}}} where ⌊ s ⌋ {\displaystyle \lfloor s\rfloor } and { s } {\displaystyle \{s\}} denote the integer part and the fractional part of s {\displaystyle s} , respectively, i.e. s = ⌊ s ⌋ + { s } {\displaystyle s=\lfloor s\rfloor +\{s\}} . [19] In this definition, there is the understanding that W 0 , p ( Ω ) = L p ( Ω ) {\displaystyle W^{0,p}(\Omega )=L^{p}(\Omega )} , so that the usual Sobolev spaces are recovered whenever s {\displaystyle s} is a positive integer. These spaces are often referred to as fractional Sobolev spaces. A generalization of the Gagliardo-Nirenberg inequality to these spaces reads
Theorem [20] (Brezis-Mironescu) — Let Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} be either the whole space, a half-space or a bounded Lipschitz domain. Let 1 ≤ p , p 1 , p 2 ≤ + ∞ {\displaystyle 1\leq p,\,p_{1},\,p_{2}\leq +\infty } be three positive extended real quantities and let s , s 1 , s 2 {\displaystyle s,\,s_{1},\,s_{2}} be non-negative real numbers. Furthermore, let θ ∈ ( 0 , 1 ) {\displaystyle \theta \in (0,1)} and assume that s 1 ≤ s 2 , s = θ s 1 + ( 1 − θ ) s 2 , 1 p = θ p 1 + 1 − θ p 2 {\displaystyle s_{1}\leq s_{2},\qquad s=\theta s_{1}+(1-\theta )s_{2},\qquad {\dfrac {1}{p}}={\dfrac {\theta }{p_{1}}}+{\dfrac {1-\theta }{p_{2}}}} hold. Then, ‖ u ‖ W s , p ( Ω ) ≤ C ‖ u ‖ W s 1 , p 1 ( Ω ) θ ‖ u ‖ W s 2 , p 2 ( Ω ) 1 − θ {\displaystyle \|u\|_{W^{s,p}(\Omega )}\leq C\|u\|_{W^{s_{1},p_{1}}(\Omega )}^{\theta }\|u\|_{W^{s_{2},p_{2}}(\Omega )}^{1-\theta }} for any u ∈ W s 1 , p 1 ( Ω ) ∩ W s 2 , p 2 ( Ω ) {\displaystyle u\in W^{s_{1},p_{1}}(\Omega )\cap W^{s_{2},p_{2}}(\Omega )} if and only if at least one of { s 2 ∈ N and s 2 ≥ 1 , p 2 = 1 , 0 < s 2 − s 1 ≤ 1 − 1 p 1 is false. {\displaystyle {\text{at least one of}}\quad {\begin{cases}s_{2}\in \mathbb {N} {\text{ and }}s_{2}\geq 1,\\p_{2}=1,\\0<s_{2}-s_{1}\leq 1-\displaystyle {\dfrac {1}{p_{1}}}\end{cases}}\quad {\text{is false.}}} The constant C > 0 {\displaystyle C>0} depends on the parameters p , p 1 , p 2 , s , s 1 , s 2 , θ {\displaystyle p,\,p_{1},\,p_{2},\,s,\,s_{1},\,s_{2},\,\theta } , on the domain Ω {\displaystyle \Omega } , but not on u {\displaystyle u} .
For example, the parameter choice p = 8 3 , p 1 = 2 , p 2 = 4 , s = 7 12 , s 1 = 1 2 , s 2 = 2 3 , θ = 1 2 {\displaystyle p={\dfrac {8}{3}},\quad p_{1}=2,\quad p_{2}=4,\quad s={\dfrac {7}{12}},\quad s_{1}={\dfrac {1}{2}},\quad s_{2}={\dfrac {2}{3}},\quad \theta ={\dfrac {1}{2}}} gives the estimate ‖ u ‖ W 7 12 , 8 3 ( Ω ) ≤ C ‖ u ‖ W 1 2 , 2 ( Ω ) 1 2 ‖ u ‖ W 2 3 , 4 ( Ω ) 1 2 . {\displaystyle \|u\|_{W^{{\frac {7}{12}},{\frac {8}{3}}}(\Omega )}\leq C\|u\|_{W^{{\frac {1}{2}},2}(\Omega )}^{\frac {1}{2}}\|u\|_{W^{{\frac {2}{3}},4}(\Omega )}^{\frac {1}{2}}.} The validity of the estimate is granted, for instance, from the fact that p 2 ≠ 1 {\displaystyle p_{2}\neq 1} .