Obstacle problem

<h2 id="historical-note">Historical note</h2>
<blockquote><p>Qualche tempo dopo Stampacchia, partendo sempre dalla sua disequazione variazionale, aperse un nuovo campo di ricerche che si rivelò importante e fecondo. Si tratta di quello che oggi è chiamato il <i>problema dell'ostacolo</i>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></p>— <a href="/facts/Sandro_Faedo/be4TP9jP">Sandro Faedo</a>, (Faedo 1986, p. 107)</blockquote>
<h2 id="motivating-problems">Motivating problems</h2>
<h3>Shape of a membrane above an obstacle</h3>
<p>The obstacle problem arises when one considers the shape taken by a soap film in a domain whose boundary position is fixed (see <a href="/facts/Plateau%2527s_problem/UyRuUh2c">Plateau's problem</a>), with the added constraint that the membrane is constrained to lie above some obstacle 
  
    
      
        ϕ
        (
        x
        )
      
    
    {\displaystyle \phi (x)}
  
 in the interior of the domain as well.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> In this case, the energy functional to be minimized is the surface area integral, or
</p>

J
          (
          u
          )
          =
          
            ∫
            
              D
            
          
          
            
              1
              +
              
                |
              
              ∇
              u
              
                
                  |
                
                
                  2
                
              
            
          
          
          
            d
          
          x
          .
        
      
    
    {\displaystyle \displaystyle J(u)=\int _{D}{\sqrt {1+|\nabla u|^{2}}}\,\mathrm {d} x.}

<p>This problem can be <i>linearized</i> in the case of small perturbations by expanding the energy functional in terms of its <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> and taking the first term only, in which case the energy to be minimized is the standard <a href="/facts/Dirichlet_energy/M31YfPKF">Dirichlet energy</a>
</p>

J
          (
          u
          )
          =
          
            ∫
            
              D
            
          
          
            |
          
          ∇
          u
          
            
              |
            
            
              2
            
          
          
            d
          
          x
          .
        
      
    
    {\displaystyle \displaystyle J(u)=\int _{D}|\nabla u|^{2}\mathrm {d} x.}

<h3>Optimal stopping</h3>
<p>The obstacle problem also arises in <a href="/facts/Control_theory/aIuERpn6">control theory</a>, specifically the question of finding the optimal stopping time for a <a href="/facts/Stochastic_process/Ng1n1dnB">stochastic process</a> with payoff function 
  
    
      
        ϕ
        (
        x
        )
      
    
    {\displaystyle \phi (x)}
  
.
</p><p>In the simple case wherein the process is <a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a>, and the process is forced to stop upon exiting the domain, the solution 
  
    
      
        u
        (
        x
        )
      
    
    {\displaystyle u(x)}
  
 of the obstacle problem can be characterized as the expected value of the payoff, starting the process at 
  
    
      
        x
      
    
    {\displaystyle x}
  
, if the optimal stopping strategy is followed. The stopping criterion is simply that one should stop upon reaching the <i>contact set</i>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="formal-statement">Formal statement</h2>
<p>Suppose the following data is given:
</p>
<ol><li>an <a href="/facts/Open_set/QfTCIqMu">open</a> <a href="/facts/Bounded_set/yCldjtoy">bounded</a> <a href="/facts/Domain_(mathematical_analysis)/xT44oI3E">domain</a> 
  
    
      
        D
        ⊆
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle D\subseteq \mathbb {R} ^{n}}
  
 with <a href="/facts/Smooth_function/mffL4ch2">smooth</a> <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a></li>
<li>a <a href="/facts/Smooth_function/mffL4ch2">smooth function</a> 
  
    
      
        f
      
    
    {\displaystyle f}
  
 on 
  
    
      
        ∂
        D
      
    
    {\displaystyle \partial D}
  
 (the <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a> of 
  
    
      
        D
      
    
    {\displaystyle D}
  
)</li>
<li>a smooth function 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  
 defined on all of 
  
    
      
        D
      
    
    {\displaystyle D}
  
 such that 
  
    
      
        φ
        
          
            |
          
          
            ∂
            D
          
        
        <
        f
      
    
    {\displaystyle \varphi |_{\partial D}<f}
  
, i.e., the restriction of 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  
 to the boundary of 
  
    
      
        D
      
    
    {\displaystyle D}
  
 (its <a href="/facts/Trace_operator/7uQ4kSE4">trace</a>) is less than 
  
    
      
        f
      
    
    {\displaystyle f}
  
.</li></ol>
<p>Then consider the set
</p>

K
          =
          
            {
            
              v
              ∈
              
                H
                
                  1
                
              
              (
              D
              )
              :
              v
              
                
                  |
                
                
                  ∂
                  D
                
              
              =
              f
              
                 and 
              
              v
              ≥
              φ
            
            }
          
          ,
        
      
    
    {\displaystyle \displaystyle K=\left\{v\in H^{1}(D):v|_{\partial D}=f{\text{ and }}v\geq \varphi \right\},}

<p>which is a <a href="/facts/Closed_set/upYnRTUj">closed</a> <a href="/facts/Convex_set/vdAuJRJl">convex</a> <a href="/facts/Subset/SJGERmbA">subset</a> of the <a href="/facts/Sobolev_space/34qxWCyr">Sobolev space</a> 
  
    
      
        
          H
          
            1
          
        
        (
        D
        )
      
    
    {\displaystyle H^{1}(D)}
  
 of square <a href="/facts/Integrable_function/V7lyV997">integrable functions</a> with domain 
  
    
      
        D
      
    
    {\displaystyle D}
  
 whose <a href="/facts/Weak_derivative/oehSmSMg">weak first derivatives</a> is square integrable, containing those functions with the desired boundary conditions and whose values above the obstacle's. A solution to the obstacle problem is a function 
  
    
      
        u
        ∈
        K
      
    
    {\displaystyle u\in K}
  
 which minimizes the energy <a href="/facts/Integral/V7lyV997">integral</a>
</p>

J
          (
          v
          )
          =
          
            ∫
            
              D
            
          
          
            |
          
          ∇
          v
          
            
              |
            
            
              2
            
          
          
            d
          
          x
        
      
    
    {\displaystyle \displaystyle J(v)=\int _{D}|\nabla v|^{2}\mathrm {d} x}

<p>over all functions 
  
    
      
        v
      
    
    {\displaystyle v}
  
 belonging to 
  
    
      
        K
      
    
    {\displaystyle K}
  
; in symbols
</p>

J
        (
        u
        )
        =
        
          min
          
            v
            ∈
            K
          
        
        ⁡
        J
        (
        v
        )
        
           or 
        
        u
        ∈
        
          Argmin
          
            K
          
        
        ⁡
        J
        .
      
    
    {\displaystyle J(u)=\operatorname {min} _{v\in K}J(v){\text{ or }}u\in \operatorname {Argmin} _{K}J.}

<p>The existence and uniqueness of such a minimizer is assured by considerations of <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> theory.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<h2 id="alternative-formulations">Alternative formulations</h2>
<h3>Variational inequality</h3>
<p class="note">See also: <a href="/facts/Variational_inequality/p3Hy60WS">Variational inequality</a></p>
<p>The obstacle problem can be reformulated as a standard problem in the theory of <a href="/facts/Variational_inequality/p3Hy60WS">variational inequalities</a> on <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a>. Seeking the energy minimizer in the set 
  
    
      
        K
      
    
    {\displaystyle K}
  
 of suitable functions is equivalent to seeking
</p>

u
          ∈
          K
        
      
    
    {\displaystyle \displaystyle u\in K}
  
    such that   
  
    
      
        
          ∫
          
            D
          
        
        
          ∇
          u
        
        ⋅
        
          ∇
          (
          v
          −
          u
          )
        
        d
        ⁡
        x
        ≥
        0
        
        ∀
        v
        ∈
        K
        ,
      
    
    {\displaystyle \int _{D}{\nabla u}\cdot {\nabla (v-u)}\operatorname {d} x\geq 0\qquad \forall v\in K,}

<p>where 
  
    
      
        ⋅
        :
        
          
            R
          
          
            n
          
        
        ×
        
          
            R
          
          
            n
          
        
        →
        
          R
        
      
    
    {\displaystyle \cdot :\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} }
  
 is the ordinary <a href="/facts/Scalar_product/tNz8MLjT">scalar product</a> in the <a href="/facts/Dimension_(mathematics)/0ktmUyac">finite-dimensional</a> <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
. This is a special case of the more general form for variational inequalities on Hilbert spaces, whose solutions are functions 
  
    
      
        u
      
    
    {\displaystyle u}
  
 in some closed convex subset 
  
    
      
        K
      
    
    {\displaystyle K}
  
 of the overall space, such that
</p>

a
          (
          u
          ,
          v
          −
          u
          )
          ≥
          l
          (
          v
          −
          u
          )
          
          ∀
          v
          ∈
          K
          .
          
        
      
    
    {\displaystyle \displaystyle a(u,v-u)\geq l(v-u)\qquad \forall v\in K.\,}

<p>for <a href="/facts/Coercive_function/6Yb5rG1q">coercive</a>, <a href="/facts/Real_numbers/R02gw5Pb">real-valued</a>, <a href="/facts/Bounded_operator/LYmDTIqR">bounded</a> <a href="/facts/Bilinear_form/gyvzn9ym">bilinear forms</a> 
  
    
      
        (
        v
        ,
        w
        )
        ↦
        a
        (
        v
        ,
        w
        )
      
    
    {\displaystyle (v,w)\mapsto a(v,w)}
  
 and bounded <a href="/facts/Linear_functional/oGh7Asrz">linear functionals</a> 
  
    
      
        v
        ↦
        l
        (
        v
        )
      
    
    {\displaystyle v\mapsto l(v)}

on 
  
    
      
        
          H
          
            1
          
        
        (
        D
        )
      
    
    {\displaystyle H^{1}(D)}
  
.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p>
<h3>Least superharmonic function</h3>
<p class="note">See also: <a href="/facts/Superharmonic_function/zleerMbw">Superharmonic function</a> and <a href="/facts/Viscosity_solution/3S8q3rUo">Viscosity solution</a></p>
<p>A variational argument shows that, away from the contact set, the solution to the obstacle problem is harmonic. A similar argument which restricts itself to variations that are positive shows that the solution is superharmonic on the contact set. Together, the two arguments imply that the solution is a superharmonic function.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
</p><p>In fact, an application of the <a href="/facts/Maximum_principle/TcomDNRa">maximum principle</a> then shows that the solution to the obstacle problem is the least superharmonic function in the set of admissible functions.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>
</p>
<h2 id="regularity-properties">Regularity properties</h2>

<h3>Optimal regularity</h3>
<p>The solution to the obstacle problem has 
  
    
      
        
          C
          
            1
            ,
            1
          
        
      
    
    {\displaystyle C^{1,1}}
  
 regularity, or <a href="/facts/Bounded_function/whVjbtuk">bounded</a> <a href="/facts/Derivative/7Ito70Dh">second derivatives</a>, when the obstacle itself has these properties.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> More precisely, the solution's <a href="/facts/Modulus_of_continuity/31lThXhW">modulus of continuity</a> and the modulus of continuity for its <a href="/facts/Derivative/7Ito70Dh">derivative</a> are related to those of the obstacle. 
</p>
<ol><li>If the obstacle 
  
    
      
        ϕ
        (
        x
        )
      
    
    {\displaystyle \phi (x)}
  
 has modulus of continuity 
  
    
      
        σ
        (
        r
        )
      
    
    {\displaystyle \sigma (r)}
  
, that is to say that 
  
    
      
        
          |
        
        ϕ
        (
        x
        )
        −
        ϕ
        (
        y
        )
        
          |
        
        ≤
        σ
        (
        
          |
        
        x
        −
        y
        
          |
        
        )
      
    
    {\displaystyle |\phi (x)-\phi (y)|\leq \sigma (|x-y|)}
  
, then the solution 
  
    
      
        u
        (
        x
        )
      
    
    {\displaystyle u(x)}
  
 has modulus of continuity given by 
  
    
      
        C
        σ
        (
        2
        r
        )
      
    
    {\displaystyle C\sigma (2r)}
  
, where the constant depends only on the domain and not the obstacle.</li>
<li>If the obstacle's first derivative has modulus of continuity 
  
    
      
        σ
        (
        r
        )
      
    
    {\displaystyle \sigma (r)}
  
, then the solution's first derivative has modulus of continuity given by 
  
    
      
        C
        r
        σ
        (
        2
        r
        )
      
    
    {\displaystyle Cr\sigma (2r)}
  
, where the constant again depends only on the domain.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a></li></ol>
<h3>Level surfaces and the free boundary</h3>
<p>Subject to a degeneracy condition, level sets of the difference between the solution and the obstacle, 
  
    
      
        {
        x
        :
        u
        (
        x
        )
        −
        ϕ
        (
        x
        )
        =
        t
        }
      
    
    {\displaystyle \{x:u(x)-\phi (x)=t\}}
  
 for 
  
    
      
        t
        >
        0
      
    
    {\displaystyle t>0}
  
 are 
  
    
      
        
          C
          
            1
            ,
            α
          
        
      
    
    {\displaystyle C^{1,\alpha }}
  
 surfaces. The free boundary, which is the boundary of the set where the solution meets the obstacle, is also 
  
    
      
        
          C
          
            1
            ,
            α
          
        
      
    
    {\displaystyle C^{1,\alpha }}
  
 except on a set of <i>singular points,</i> which are themselves either isolated or locally contained on a 
  
    
      
        
          C
          
            1
          
        
      
    
    {\displaystyle C^{1}}
  
 manifold.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a>
</p>
<h2 id="generalizations">Generalizations</h2>
<p>The theory of the obstacle problem is extended to other divergence form uniformly <a href="/facts/Elliptic_operator/3lLoapry">elliptic operators</a>,<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> and their associated energy functionals. It can be generalized to degenerate elliptic operators as well.
</p><p>The double obstacle problem, where the function is constrained to lie above one obstacle function and below another, is also of interest.
</p><p>The <a href="/facts/Signorini_problem/8M2mnxuK">Signorini problem</a> is a variant of the obstacle problem, where the energy functional is minimized subject to a constraint which only lives on a surface of one lesser dimension, which includes the <i>boundary obstacle problem</i>, where the constraint operates on the boundary of the domain.
</p><p>The <a href="/facts/Parabolic_partial_differential_equation/gJQfRUlr">parabolic</a>, time-dependent cases of the obstacle problem and its variants are also objects of study.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Minimal_surface/Fy7VkrLQ">Minimal surface</a></li>
<li><a href="/facts/Variational_inequality/p3Hy60WS">Variational inequality</a></li>
<li><a href="/facts/Signorini_problem/8M2mnxuK">Signorini problem</a></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="historical-references">Historical references</h2>
<ul><li><a href="/facts/Sandro_Faedo/be4TP9jP">Faedo, Sandro</a> (1986), "Leonida Tonelli e la scuola matematica pisana", in Montalenti, G.; <a href="/facts/Luigi_Amerio/Iol2Mh7t">Amerio, L.</a>; Acquaro, G.; Baiada, E.; et al. (eds.), <a href="https://web.archive.org/web/20110223030014/http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=32847"><i>Convegno celebrativo del centenario della nascita di Mauro Picone e Leonida Tonelli (6–9 maggio 1985)</i></a>, Atti dei Convegni Lincei (in Italian), vol. 77, Roma: <a href="/facts/Accademia_Nazionale_dei_Lincei/XjuCP82q">Accademia Nazionale dei Lincei</a>, pp. 89–109, archived from <a href="http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=32847">the original</a> on 2011-02-23, retrieved 2013-02-12. "<i>Leonida Tonelli and the Pisa mathematical school</i>" is a survey of the work of Tonelli in <a href="/facts/Pisa/AghIeTCI">Pisa</a> and his influence on the development of the school, presented at the <i>International congress in occasion of the celebration of the centenary of birth of Mauro Picone and Leonida Tonelli</i> (held in <a href="/facts/Rome/bh1Ttx3T">Rome</a> on May 6–9, 1985). The Author was one of his pupils and, after his death, held his chair of mathematical analysis at the <a href="/facts/University_of_Pisa/EXvLLKQr">University of Pisa</a>, becoming dean of the faculty of sciences and then rector: he exerted a strong positive influence on the development of the university.</li></ul>

<ul><li><a href="/facts/Luis_Caffarelli/9Yulcg1F">Caffarelli, Luis</a> (July 1998), "The obstacle problem revisited", <i>The Journal of Fourier Analysis and Applications</i>, 4 (4–5): 383–402, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1998JFAA....4..383C">1998JFAA....4..383C</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02498216">10.1007/BF02498216</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1658612">1658612</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123431389">123431389</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0928.49030">0928.49030</a></li>
<li><a href="/facts/Lawrence_C._Evans/0XluMrne">Evans, Lawrence</a>, <a href="http://math.berkeley.edu/~evans/SDE.course.pdf"><i>An Introduction to Stochastic Differential Equations</i></a> (PDF), p. 130, retrieved July 11, 2011. A set of lecture notes surveying "<i>without too many precise details, the basic theory of probability, random differential equations and some applications</i>", as the author himself states.</li>
<li>Frehse, Jens (1972), "On the regularity of the solution of a second order variational inequality", <i>Bolletino della Unione Matematica Italiana</i>, Serie IV, vol. 6, pp. 312–315, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0318650">0318650</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0261.49021">0261.49021</a>.</li>
<li><a href="/facts/Avner_Friedman/b0KkPJFq">Friedman, Avner</a> (1982), <i>Variational principles and free boundary problems</i>, Pure and Applied Mathematics, New York: <a href="/facts/John_Wiley_%2526_Sons/53g3LqJc">John Wiley & Sons</a>, pp. ix+710, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-86849-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0679313">0679313</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0564.49002">0564.49002</a>.</li>
<li><a href="/facts/David_Kinderlehrer/YEBdCFbo">Kinderlehrer, David</a>; <a href="/facts/Guido_Stampacchia/Gw5ATqXt">Stampacchia, Guido</a> (1980), <i>An Introduction to Variational Inequalities and Their Applications</i>, Pure and Applied Mathematics, vol. 88, New York: <a href="/facts/Academic_Press/qxoKjzFy">Academic Press</a>, pp. xiv+313, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-407350-6, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0567696">0567696</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0457.35001">0457.35001</a></li>
<li>Petrosyan, Arshak; Shahgholian, Henrik; Uraltseva, Nina (2012), <i>Regularity of Free Boundaries in Obstacle-Type Problems. Graduate Studies in Mathematics</i>, American Mathematical Society, Providence, RI, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-8794-3</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Luis_Caffarelli/9Yulcg1F">Caffarelli, Luis</a> (August 1998), <a href="http://www.ma.utexas.edu/users/combs/obstacle-long.pdf"><i>The Obstacle Problem</i></a> (PDF), draft from the Fermi Lectures, p. 45, retrieved July 11, 2011, delivered by the author at the <a href="/facts/Scuola_Normale_Superiore/Qhzq6LEW">Scuola Normale Superiore</a> in 1998.</li></ul>

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

<ol>
<li id="fn:1"><p>See Caffarelli 1998, p. 384. - Caffarelli, Luis (July 1998), "The obstacle problem revisited", The Journal of Fourier Analysis and Applications, 4 (4–5): 383–402, Bibcode:1998JFAA....4..383C, doi:10.1007/BF02498216, MR 1658612, S2CID 123431389, Zbl 0928.49030 <a href="https://ui.adsabs.harvard.edu/abs/1998JFAA....4..383C" target="_blank">https://ui.adsabs.harvard.edu/abs/1998JFAA....4..383C</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"Some time after Stampacchia, starting again from his variational inequality, opened a new field of research, which revealed itself as important and fruitful. It is the now called obstacle problem" (English translation). The Italic type emphasis is due to the author himself. <a href="/wiki/Italic_type" target="_blank">/wiki/Italic_type</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>See Caffarelli 1998, p. 383. - Caffarelli, Luis (July 1998), "The obstacle problem revisited", The Journal of Fourier Analysis and Applications, 4 (4–5): 383–402, Bibcode:1998JFAA....4..383C, doi:10.1007/BF02498216, MR 1658612, S2CID 123431389, Zbl 0928.49030 <a href="https://ui.adsabs.harvard.edu/abs/1998JFAA....4..383C" target="_blank">https://ui.adsabs.harvard.edu/abs/1998JFAA....4..383C</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>See the lecture notes by Evans, pp. 110–114). - Evans, Lawrence, An Introduction to Stochastic Differential Equations (PDF), p. 130, retrieved July 11, 2011 <a href="http://math.berkeley.edu/~evans/SDE.course.pdf" target="_blank">http://math.berkeley.edu/~evans/SDE.course.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>See Caffarelli 1998, p. 383. - Caffarelli, Luis (July 1998), "The obstacle problem revisited", The Journal of Fourier Analysis and Applications, 4 (4–5): 383–402, Bibcode:1998JFAA....4..383C, doi:10.1007/BF02498216, MR 1658612, S2CID 123431389, Zbl 0928.49030 <a href="https://ui.adsabs.harvard.edu/abs/1998JFAA....4..383C" target="_blank">https://ui.adsabs.harvard.edu/abs/1998JFAA....4..383C</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>See Kinderlehrer & Stampacchia 1980, pp. 40–41. - Kinderlehrer, David; Stampacchia, Guido (1980), An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics, vol. 88, New York: Academic Press, pp. xiv+313, ISBN 0-12-407350-6, MR 0567696, Zbl 0457.35001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0567696" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0567696</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>See Kinderlehrer & Stampacchia 1980, pp. 23–49. - Kinderlehrer, David; Stampacchia, Guido (1980), An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics, vol. 88, New York: Academic Press, pp. xiv+313, ISBN 0-12-407350-6, MR 0567696, Zbl 0457.35001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0567696" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0567696</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>See Caffarelli 1998, p. 384. - Caffarelli, Luis (July 1998), "The obstacle problem revisited", The Journal of Fourier Analysis and Applications, 4 (4–5): 383–402, Bibcode:1998JFAA....4..383C, doi:10.1007/BF02498216, MR 1658612, S2CID 123431389, Zbl 0928.49030 <a href="https://ui.adsabs.harvard.edu/abs/1998JFAA....4..383C" target="_blank">https://ui.adsabs.harvard.edu/abs/1998JFAA....4..383C</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>See Kinderlehrer & Stampacchia 1980, pp. 23–49. - Kinderlehrer, David; Stampacchia, Guido (1980), An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics, vol. 88, New York: Academic Press, pp. xiv+313, ISBN 0-12-407350-6, MR 0567696, Zbl 0457.35001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0567696" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0567696</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>See Frehse 1972. - Frehse, Jens (1972), "On the regularity of the solution of a second order variational inequality", Bolletino della Unione Matematica Italiana, Serie IV, vol. 6, pp. 312–315, MR 0318650, Zbl 0261.49021 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0318650" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0318650</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>See Caffarelli 1998, p. 386. - Caffarelli, Luis (July 1998), "The obstacle problem revisited", The Journal of Fourier Analysis and Applications, 4 (4–5): 383–402, Bibcode:1998JFAA....4..383C, doi:10.1007/BF02498216, MR 1658612, S2CID 123431389, Zbl 0928.49030 <a href="https://ui.adsabs.harvard.edu/abs/1998JFAA....4..383C" target="_blank">https://ui.adsabs.harvard.edu/abs/1998JFAA....4..383C</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>See Caffarelli 1998, pp. 394 and 397. - Caffarelli, Luis (July 1998), "The obstacle problem revisited", The Journal of Fourier Analysis and Applications, 4 (4–5): 383–402, Bibcode:1998JFAA....4..383C, doi:10.1007/BF02498216, MR 1658612, S2CID 123431389, Zbl 0928.49030 <a href="https://ui.adsabs.harvard.edu/abs/1998JFAA....4..383C" target="_blank">https://ui.adsabs.harvard.edu/abs/1998JFAA....4..383C</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>See Kinderlehrer & Stampacchia 1980, pp. 23–49. - Kinderlehrer, David; Stampacchia, Guido (1980), An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics, vol. 88, New York: Academic Press, pp. xiv+313, ISBN 0-12-407350-6, MR 0567696, Zbl 0457.35001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0567696" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0567696</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
</ol>

Obstacle problem open-in-new

Obstacle problem