Regularity theory

<h2 id="elliptic-regularity-theory">Elliptic regularity theory</h2>
<p class="note">Main article: <a href="/facts/Elliptic_boundary_value_problem/oxW9cVDd">Elliptic boundary value problem</a></p>
<p>Let 
  
    
      
        U
      
    
    {\displaystyle U}
  
 be an <a href="/facts/Open_set/QfTCIqMu">open</a>, <a href="/facts/Bounded_set/yCldjtoy">bounded</a> subset of 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
, denote its boundary as 
  
    
      
        ∂
        U
      
    
    {\displaystyle \partial U}
  
 and the variables as 
  
    
      
        x
        =
        (
        
          x
          
            1
          
        
        ,
        .
        .
        .
        ,
        
          x
          
            n
          
        
        )
      
    
    {\displaystyle x=(x_{1},...,x_{n})}
  
. Representing the PDE as a partial <a href="/facts/Differential_operator/Nr15J0IE">differential operator</a> 
  
    
      
        L
      
    
    {\displaystyle L}
  
 acting on an unknown function 
  
    
      
        u
        =
        u
        (
        x
        )
      
    
    {\displaystyle u=u(x)}
  
 of 
  
    
      
        x
        ∈
        U
      
    
    {\displaystyle x\in U}
  
 results in a BVP of the form 
  
    
      
        
          {
          
            
              
                
                  L
                  u
                
                
                  
                  =
                  f
                
                
                
                  
                    in 
                  
                  U
                
              
              
                
                  u
                
                
                  
                  =
                  0
                
                
                
                  
                    on 
                  
                  ∂
                  U
                  ,
                
              
            
          
          
        
      
    
    {\displaystyle \left\{{\begin{aligned}Lu&=f&&{\text{in }}U\\u&=0&&{\text{on }}\partial U,\end{aligned}}\right.}
  
 where 
  
    
      
        f
        :
        U
        →
        
          R
        
      
    
    {\displaystyle f:U\rightarrow \mathbb {R} }
  
 is a given function 
  
    
      
        f
        =
        f
        (
        x
        )
      
    
    {\displaystyle f=f(x)}
  
 and 
  
    
      
        u
        :
        U
        ∪
        ∂
        U
        →
        
          R
        
      
    
    {\displaystyle u:U\cup \partial U\rightarrow \mathbb {R} }
  
 and the <a href="/facts/Elliptic_operator/3lLoapry">elliptic operator</a> 
  
    
      
        L
      
    
    {\displaystyle L}
  
 is of the divergence form: 
  
    
      
        L
        u
        (
        x
        )
        =
        −
        
          ∑
          
            i
            ,
            j
            =
            1
          
          
            n
          
        
        (
        
          a
          
            i
            j
          
        
        (
        x
        )
        
          u
          
            
              x
              
                i
              
            
          
        
        
          )
          
            
              x
              
                j
              
            
          
        
        +
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        
          b
          
            i
          
        
        (
        x
        )
        
          u
          
            
              x
              
                i
              
            
          
        
        (
        x
        )
        +
        c
        (
        x
        )
        u
        (
        x
        )
        ,
      
    
    {\displaystyle Lu(x)=-\sum _{i,j=1}^{n}(a_{ij}(x)u_{x_{i}})_{x_{j}}+\sum _{i=1}^{n}b_{i}(x)u_{x_{i}}(x)+c(x)u(x),}
  
then
</p>
<ul><li>Interior regularity: If <i>m</i> is a natural number, 
  
    
      
        
          a
          
            i
            j
          
        
        ,
        
          b
          
            j
          
        
        ,
        c
        ∈
        
          C
          
            m
            +
            1
          
        
        (
        U
        )
        ,
        f
        ∈
        
          H
          
            m
          
        
        (
        U
        )
      
    
    {\displaystyle a^{ij},b^{j},c\in C^{m+1}(U),f\in H^{m}(U)}
  
 (2) , 
  
    
      
        u
        ∈
        
          H
          
            0
          
          
            1
          
        
        (
        U
        )
      
    
    {\displaystyle u\in H_{0}^{1}(U)}
  
 is a weak solution, then for any open set <i>V</i> in <i>U</i> with compact closure, 
  
    
      
        ‖
        u
        
          ‖
          
            
              H
              
                m
                +
                2
              
            
            (
            V
            )
          
        
        ≤
        C
        (
        ‖
        f
        
          ‖
          
            
              H
              
                m
              
            
            (
            U
            )
          
        
        +
        ‖
        u
        
          ‖
          
            
              L
              
                2
              
            
            (
            U
            )
          
        
        )
      
    
    {\displaystyle \|u\|_{H^{m+2}(V)}\leq C(\|f\|_{H^{m}(U)}+\|u\|_{L^{2}(U)})}
  
(3), where <i>C</i> depends on <i>U, V, L, m</i>, per se 
  
    
      
        u
        ∈
        
          H
          
            l
            o
            c
          
          
            m
            +
            2
          
        
        (
        U
        )
      
    
    {\displaystyle u\in H_{loc}^{m+2}(U)}
  
, which also holds if <i>m</i> is infinity by <a href="/facts/Sobolev_inequality/jxJvCHML">Sobolev embedding theorem</a>.</li>
<li>Boundary regularity: (2) together with the assumption that 
  
    
      
        ∂
        U
      
    
    {\displaystyle \partial U}
  
 is 
  
    
      
        
          C
          
            m
            +
            2
          
        
      
    
    {\displaystyle C^{m+2}}
  
 indicates that (3) still holds after replacing <i>V</i> with <i>U,</i> i.e. 
  
    
      
        u
        ∈
        
          H
          
            m
            +
            2
          
        
        (
        U
        )
      
    
    {\displaystyle u\in H^{m+2}(U)}
  
, which also holds if <i>m</i> is infinity.</li></ul>
<h2 id="parabolic-and-hyperbolic-regularity-theory">Parabolic and Hyperbolic regularity theory</h2>
<p><a href="/facts/Parabolic_partial_differential_equation/gJQfRUlr">Parabolic</a> and <a href="/facts/Hyperbolic_partial_differential_equation/znXgGuvh">hyperbolic</a> PDEs describe the time evolution of a quantity <i>u</i> governed by an <a href="/facts/Elliptic_operator/3lLoapry">elliptic operator</a> <i>L</i> and an external force <i>f</i> over a space 
  
    
      
        U
        ⊂
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle U\subset \mathbb {R} ^{n}}
  
. We assume the boundary of <i>U</i> to be smooth, and the elliptic operator to be independent of time, with smooth coefficients, i.e.
  
    
      
        L
        u
        (
        t
        ,
        x
        )
        =
        −
        
          ∑
          
            i
            ,
            j
            =
            1
          
          
            n
          
        
        
          
            (
          
        
        
          a
          
            i
            j
          
        
        (
        x
        )
        
          u
          
            
              x
              
                i
              
            
          
        
        (
        t
        ,
        x
        )
        
          
            
              )
            
          
          
            
              x
              
                j
              
            
          
        
        +
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        
          b
          
            i
          
        
        (
        x
        )
        
          u
          
            
              x
              
                i
              
            
          
        
        (
        t
        ,
        x
        )
        +
        c
        (
        x
        )
        u
        (
        t
        ,
        x
        )
        .
      
    
    {\displaystyle Lu(t,x)=-\sum _{i,j=1}^{n}{\big (}a_{ij}(x)u_{x_{i}}(t,x){\big )}_{x_{j}}+\sum _{i=1}^{n}b_{i}(x)u_{x_{i}}(t,x)+c(x)u(t,x).}
  
In addition, we subscribe the boundary value of <i>u</i> to be 0.
</p><p>Then the regularity of the solution is given by the following table,
</p>
<table><tbody><tr><th>Equation</th><th>                              u                      t                          +        L        u        =        f              {\displaystyle u_{t}+Lu=f}   (parabolic)</th><th>                              u                      t            t                          +        L        u        =        f              {\displaystyle u_{tt}+Lu=f}   (hyperbolic)</th></tr><tr><td>Initial Condition</td><td>                    u        (        0        )        ∈                  H                      x                                2            m            +            1                                {\displaystyle u(0)\in H_{x}^{2m+1}}  </td><td>                    u        (        0        )        ∈                  H                      x                                m            +            1                          ,                (                  ∂                      t                          u        )        (        0        )        ∈                  H                      x                                m                                {\displaystyle u(0)\in H_{x}^{m+1},\,(\partial _{t}u)(0)\in H_{x}^{m}}  </td></tr><tr><td>External force</td><td>                              ∂                      t                                k                          f        ∈                  L                      t                                2                                    H                      x                                2            (            m            −            k            )                                  (        k        =        1        ,        …        m        )              {\displaystyle \partial _{t}^{k}f\in L_{t}^{2}H_{x}^{2(m-k)}\,(k=1,\dots m)}  </td><td>                              ∂                      t                                k                          f        ∈                  L                      t                                2                                    H                      x                                m            −            k                                  (        k        =        1        ,        …        m        )              {\displaystyle \partial _{t}^{k}f\in L_{t}^{2}H_{x}^{m-k}\,(k=1,\dots m)}  </td></tr><tr><td>Solution</td><td>                              ∂                      t                                k                          u        ∈                  L                      t                                2                                    H                      x                                2            (            m            +            1            −            k            )                          ,                (        k        =        1        ,        …        ,        m        +        1        )              {\displaystyle \partial _{t}^{k}u\in L_{t}^{2}H_{x}^{2(m+1-k)},\,(k=1,\dots ,m+1)}  </td><td>                              ∂                      t                                k                          u        ∈                  L                      t                                ∞                                    H                      x                                m            +            1            −            k                          ,                (        k        =        1        ,        …        ,        m        +        1        )              {\displaystyle \partial _{t}^{k}u\in L_{t}^{\infty }H_{x}^{m+1-k},\,(k=1,\dots ,m+1)}  </td></tr></tbody></table>
<p>where <i>m</i> is a natural number, 
  
    
      
        x
        ∈
        U
      
    
    {\displaystyle x\in U}
  
 denotes the space variable, <i>t</i> denotes the time variable, <i>Hs</i> is a <a href="/facts/Sobolev_space/34qxWCyr">Sobolev space</a> of functions with square-integrable weak derivatives, and <i>LtpX</i> is the <a href="/facts/Bochner_space/tSuPabsJ">Bochner space</a> of integrable <i>X</i>-valued functions.
</p>
<h2 id="counterexamples">Counterexamples</h2>
<p>Not every weak solution is smooth; for example, there may be discontinuities in the weak solutions of <a href="/facts/Conservation_law/6g1Ay2FR">conservation laws</a> called shock waves.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Fernández-Real, Xavier; Ros-Oton, Xavier (2022-12-06). Regularity Theory for Elliptic PDE. arXiv:2301.01564. doi:10.4171/ZLAM/28. ISBN 978-3-98547-028-0. S2CID 254389061. <a href="978-3-98547-028-0" target="_blank">978-3-98547-028-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Evans, Lawrence C. (1998). Partial differential equations (PDF). Providence (R. I.): American mathematical society. ISBN 0-8218-0772-2. <a href="0-8218-0772-2" target="_blank">0-8218-0772-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Smoller, Joel. Shock Waves and Reaction—Diffusion Equations (2 ed.). Springer New York, NY. doi:10.1007/978-1-4612-0873-0. ISBN 978-0-387-94259-9. <a href="978-0-387-94259-9" target="_blank">978-0-387-94259-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Regularity theory open-in-new

Regularity theory