View source for Bitsadze-Lavrent'ev problem

''Tricomi–Bitsadze–Lavrent'ev problem''

The problem of finding a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105901.png" /> which satisfies

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105902.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

in a mixed domain that is simply connected and bounded by a Jordan (non-self-intersecting)  "elliptic"  arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105903.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105904.png" />) with end-points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105906.png" /> and by the  "real"  characteristics (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105907.png" />)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105908.png" /></td> </tr></table>

of the Bitsadze–Lavrent'ev equation (a1), which satisfy the characteristic equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105909.png" /></td> </tr></table>

and meet at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059010.png" />, and which assumes prescribed continuous boundary values

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059012.png" /> is the arc length reckoned from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059013.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059014.png" /></td> </tr></table>

Consider the aforementioned domain (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059015.png" />). Then a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059016.png" /> is a regular solution of the Bitsadze–Lavrent'ev problem if it satisfies the following conditions:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059017.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059018.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059020.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059021.png" /> are continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059022.png" /> (except, possibly, at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059024.png" />, where they may have poles of order less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059025.png" />, i.e., they may tend to infinity with order less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059026.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059028.png" />);

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059030.png" /> are continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059031.png" /> (except possibly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059032.png" />, where they need not exist);

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059033.png" /> satisfies (a1) at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059034.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059035.png" /> without <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059036.png" />);

5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059037.png" /> satisfies the boundary conditions (a2).

Consider the normal curve (of Bitsadze–Lavrent'ev)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059038.png" /></td> </tr></table>

Note that it is the upper semi-circle and can also be given by (the upper part of)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059039.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059040.png" />. The curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059041.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059042.png" /> in its interior.

The idea of A.V. Bitsadze and M.A. Lavrent'ev for finding regular solutions of the above problem is as follows. First, solve the problem N (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059044.png" />). That is, find a regular solution of equation (a1) satisfying the boundary conditions:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059045.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059046.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059049.png" /> is continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059051.png" />, and may tend to infinity of order less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059052.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059054.png" />.

Secondly, solve the Cauchy–Goursat problem (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059056.png" />). That is, find a regular solution of (a1) satisfying the boundary conditions:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059057.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059058.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059059.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059061.png" /> is continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059063.png" />, and may tend to infinity of order less that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059064.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059066.png" />.

Finally, take into account the boundary condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059067.png" /></td> </tr></table>

Therefore, one has a [[Goursat problem|Goursat problem]] (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059069.png" />) for (a1) with boundary conditions:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059070.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059071.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059072.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059073.png" />.

Several extensions and generalizations of the above boundary value problem of mixed type have been established [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]]. These problems are important in fluid mechanics (aerodynamics and hydrodynamics, [[#References|[a1]]], [[#References|[a2]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Bitsadze,   "Equations of mixed type" , Macmillan  (1964)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Ferrari,   F.G. Tricomi,   "Transonic aerodynamics" , Acad. Press  (1968)  (Translated from Italian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.M. Rassias,   "Mixed type equations" , '''90''' , Teubner  (1986)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.M. Rassias,   "Lecture notes on mixed type partial differential equations" , World Sci.  (1990)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.M. Rassias,   "The Bitsadze–Lavrentjev problem"  ''Bull. Soc. Roy. Sci. Liège'' , '''48'''  (1979)  pp. 424–425</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.M. Rassias,   "The bi-hyperbolic Bitsadze–Lavrentjev–Rassias problem in three-dimensional Euclidean space"  ''C.R. Acad. Sci. Bulg. Sci.'' , '''39'''  (1986)  pp. 29–32</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.M. Rassias,   "The mixed Bitsadze–Lavrentjev–Tricomi boundary value problem" , ''Texte zur Mathematik'' , '''90''' , Teubner  (1986)  pp. 6–21</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.M. Rassias,   "The well posed Tricomi–Bitsadze–Lavrentjev problem in the Euclidean plane"  ''Atti. Accad. Sci. Torino'' , '''124'''  (1990)  pp. 73–83</TD></TR></table>