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Difference between revisions of "Well-posed problem"

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The problem of determining a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w0976601.png" /> in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w0976602.png" /> (with distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w0976603.png" />) from initial data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w0976604.png" /> in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w0976605.png" /> (with distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w0976606.png" />), satisfying the following conditions: a) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w0976607.png" /> there exists a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w0976608.png" />; b) the solution is uniquely defined; c) the problem is stable with respect to the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w0976609.png" />: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w09766010.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w09766011.png" /> such that, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w09766012.png" />, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w09766013.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w09766014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w09766015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097660/w09766016.png" />.
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The problem of determining a solution $z=R(u)$ in a metric space $Z$ (with distance $\rho_Z({\cdot},{\cdot})$) from initial data $u$ in a metric space $U$ (with distance $\rho_U({\cdot},{\cdot})$), satisfying the following conditions: a) for any $u \in U$ there exists a solution $z \in Z$; b) the solution is uniquely defined; c) the problem is stable with respect to the spaces $(Z,U)$: For any $\epsilon>0$ there exists a $\delta(\epsilon)>0$ such that, for any $u_1,u_2 \in U$, the inequality $\rho_U(u_1,u_2) < \delta(\epsilon)$ implies $\rho_Z(z_1,z_2) < \epsilon$, where $z_1 = R(u_1)$, $z_2 = R(u_2)$.
  
Problems not satisfying one of these conditions for well-posedness are called [[Ill-posed problems|ill-posed problems]].
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Problems not satisfying one of these conditions for well-posedness are called [[ill-posed problems]].
  
 
====Comments====
 
====Comments====
The term  "well-posed"  (also properly posed or correctly set) was coined by the French mathematician J. Hadamard at the beginning of the 19th century [[#References|[a1]]]. In particular, he stressed the importance of continuous dependence of solutions on the data (i.e. property 3). Practical problems (e.g. in hydrodynamics, seismology) lead not seldom to formulations that are ill-posed (cf. [[Ill-posed problems|Ill-posed problems]]).
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The term  "well-posed"  (also properly posed or correctly set) was coined by the French mathematician J. Hadamard at the beginning of the 19th century [[#References|[a1]]]. In particular, he stressed the importance of continuous dependence of solutions on the data (i.e. property 3). Practical problems (e.g. in hydrodynamics, seismology) lead not seldom to formulations that are ill-posed (cf. [[Ill-posed problems]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Hadamard,  "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint  (1952)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Garabedian,  "Partial differential equations" , Wiley  (1964)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Hadamard,  "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint  (1952)  (Translated from French) {{ZBL|0049.34805}}</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Garabedian,  "Partial differential equations" , Wiley  (1964) {{ZBL|0124.30501}}</TD></TR>
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</table>
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Latest revision as of 18:55, 2 March 2018

The problem of determining a solution $z=R(u)$ in a metric space $Z$ (with distance $\rho_Z({\cdot},{\cdot})$) from initial data $u$ in a metric space $U$ (with distance $\rho_U({\cdot},{\cdot})$), satisfying the following conditions: a) for any $u \in U$ there exists a solution $z \in Z$; b) the solution is uniquely defined; c) the problem is stable with respect to the spaces $(Z,U)$: For any $\epsilon>0$ there exists a $\delta(\epsilon)>0$ such that, for any $u_1,u_2 \in U$, the inequality $\rho_U(u_1,u_2) < \delta(\epsilon)$ implies $\rho_Z(z_1,z_2) < \epsilon$, where $z_1 = R(u_1)$, $z_2 = R(u_2)$.

Problems not satisfying one of these conditions for well-posedness are called ill-posed problems.

Comments

The term "well-posed" (also properly posed or correctly set) was coined by the French mathematician J. Hadamard at the beginning of the 19th century [a1]. In particular, he stressed the importance of continuous dependence of solutions on the data (i.e. property 3). Practical problems (e.g. in hydrodynamics, seismology) lead not seldom to formulations that are ill-posed (cf. Ill-posed problems).

References

[a1] J. Hadamard, "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint (1952) (Translated from French) Zbl 0049.34805
[a2] P.R. Garabedian, "Partial differential equations" , Wiley (1964) Zbl 0124.30501
How to Cite This Entry:
Well-posed problem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Well-posed_problem&oldid=14592