Hopf maximum principle

<h2 id="mathematical-formulation">Mathematical formulation</h2>
Let u = u(x), x = (x1, ..., xn) be a C2 function which satisfies the differential inequality

L
        u
        =
        
          ∑
          
            i
            j
          
        
        
          a
          
            i
            j
          
        
        (
        x
        )
        
          
            
              
                ∂
                
                  2
                
              
              u
            
            
              ∂
              
                x
                
                  i
                
              
              ∂
              
                x
                
                  j
                
              
            
          
        
        +
        
          ∑
          
            i
          
        
        
          b
          
            i
          
        
        (
        x
        )
        
          
            
              ∂
              u
            
            
              ∂
              
                x
                
                  i
                
              
            
          
        
        ≥
        0
      
    
    {\displaystyle Lu=\sum _{ij}a_{ij}(x){\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}+\sum _{i}b_{i}(x){\frac {\partial u}{\partial x_{i}}}\geq 0}

in an <a href="/facts/Open_set/QfTCIqMu">open domain</a> (connected open subset of Rn) Ω, where the <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric matrix</a> aij = aji(x) is locally uniformly <a href="/facts/Positive_definite_matrix/Hbr6AuPS">positive definite</a> in Ω and the coefficients aij, bi are locally <a href="/facts/Bounded_set/yCldjtoy">bounded</a>. If u takes a maximum value M in Ω then u ≡ M.
The coefficients aij, bi are just functions. If they are known to be continuous then it is sufficient to demand pointwise positive definiteness of aij on the domain.
It is usually thought that the Hopf maximum principle applies only to <a href="/facts/Linear_differential_operator/Nr15J0IE">linear differential operators</a> L. In particular, this is the point of view taken by <a href="/facts/Richard_Courant/QDToNfWl">Courant</a> and <a href="/facts/David_Hilbert/1vhlHn4b">Hilbert's</a> <a href="/facts/Methoden_der_mathematischen_Physik/4tFyAK8a">Methoden der mathematischen Physik</a>. In the later sections of his original paper, however, Hopf considered a more general situation which permits certain nonlinear operators L and, in some cases, leads to uniqueness statements in the <a href="/facts/Dirichlet_problem/uHErgNCb">Dirichlet problem</a> for the <a href="/facts/Mean_curvature/lgdJ9b0k">mean curvature</a> operator and the <a href="/facts/Monge%25E2%2580%2593Amp%25C3%25A8re_equation/65CNEUGA">Monge–Ampère equation</a>.

<h2 id="boundary-behaviour">Boundary behaviour</h2>
If the domain 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 has the interior sphere property (for example, if 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 has a smooth boundary), slightly more can be said. If in addition to the assumptions above, 
 
 
 
 u
 ∈
 
 C
 
 1
 
 
 (
 
 
 Ω
 ¯
 
 
 )
 
 
 {\displaystyle u\in C^{1}({\overline {\Omega }})}
 
 and u takes a maximum value M at a point x0 in 
 
 
 
 ∂
 Ω
 
 
 {\displaystyle \partial \Omega }
 
, then for any outward direction ν at x0, there holds 
 
 
 
 
 
 
 ∂
 u
 
 
 ∂
 ν
 
 
 
 (
 
 x
 
 0
 
 
 )
 >
 0
 
 
 {\displaystyle {\frac {\partial u}{\partial \nu }}(x_{0})>0}
 
 unless 
 
 
 
 u
 ≡
 M
 
 
 {\displaystyle u\equiv M}
 
.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

<ul><li>Hopf, Eberhard (2002), Morawetz, Cathleen S.; Serrin, James B.; Sinai, Yakov G. (eds.), Selected works of Eberhard Hopf with commentaries, Providence, RI: American Mathematical Society, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-2077-X, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1985954">1985954</a>.</li>
<li>Pucci, Patrizia; Serrin, James (2004), "The strong maximum principle revisited", Journal of Differential Equations, 196 (1): 1–66, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2004JDE...196....1P">2004JDE...196....1P</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.jde.2003.05.001">10.1016/j.jde.2003.05.001</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2025185">2025185</a>.</li></ul>

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

<ol>
<li id="fn:1">Han, Qing; Lin, Fanghua (2011). Elliptic Partial Differential Equations. American Mathematical Soc. p. 28. ISBN 9780821853139. <a href="9780821853139" target="_blank">9780821853139</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Hopf maximum principle open-in-new

Hopf maximum principle